To be sure, ranging from the various Turing Machines (TM)—u-TM, o-TM, d-TM, and many others, up to the various levels of life, say, from bacteria to living cells to plants to animals, etc., there is a growing process by which new and previously unforeseen degrees of freedom are reached. Reality, nature and the world can be seen as an increasing process of degrees of

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

freedom. This view, however, should not be understood hierarchically. Rather, the architecture is to be seen as a fractal organization. As the growing architecture of life increases, an amplification of previous lower layers is produced, that does not obliterate or supersedes the lower levels.

Matter ∊ Energy ∊ Information (1)

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 85

Meaning that energy explains better and deeper what matter is supposed to explain, and that information explains still much better and deeper what energy was supposed to explain. Thus, the story from classical mechanics to thermodynamics to information theory is the very story by which a higher and more achieved understanding of reality is accomplished. This, however should not imply by any means a linear and necessarily accumulative process, but a story of improving, gaining more knowledge and wisdom about the universe, nature, and society. The story of knowledge can very appropriately be grasped in terms of

The amount of information of a system depends on how probable an event is. In other words, the higher the probability that an event takes place, the lower the amount—and the quality—of the system. And vice versa: the lower the probability that an event happens, the higher—and better—the information of that event. Briefly said, the information is inversely proportional to the probability—of an event. This means that information is a measure of how

As it can be seen, the characterization I have provided about information does not necessarily coincide with the standard view—say, Shannon- that identifies information and entropy. Rather, entropy can be adequately grasped as the quantification of randomness—of a system. Therefore, it can be easily stated that the amount of information in the universe as a whole can only increase, which means that what was considered as an expanding universe in terms of the transition from energy to matter is really an expansion of information [4]. This coincides with Formula (1) presented above. Information is the concept or reality that encompasses and explains energy and matter (mass). Whereas matter presents a view of the world rather disconnected (indeed, it is the laws that gather and unify matter, according to Newton), and energy unfolds and transforms as a set of expressions and processes explained in terms of Fourier's and Boltzmann's laws, information set out a convergent, yet in-process, reality that allows for evolution [5]. Formula (2) expresses the unity of matter, energy, and information, showing hence that there are not three things but only one—read thereafter in terms of infor-

Matter − Energy − Information (2)

Formula (2) serves as a synthesis that claims that matter, energy, and information are one and the same thing that were historically discovered or brought out by stages. That story is exactly the story that leads from classical mechanics to thermodynamics to information theory.

Now, summarized, information theory is cryptography, quantum information, entanglement, and teleportation [6]. These are three different faces of one and the same development—an achievement that crosses the most conspicuous technologies nowadays, as well as spearhead science in any domain you wish. By and large information pervades both science and culture,

bifurcations [2].

surprising something is [3].

mation, according to Formula (1).

currently.

To be sure, the process by which the living systems gain degrees of freedom coincides with the entire weave of networks, fractal architectures, and interdependencies among layers, planes, and contexts. As a result, life and nature can be seen as a fantastic interweaving of times, layers, and interdependencies.

A number of works have been done that permit the approach of this text. The various authors, takes, and approaches will be discussed along the text, which brings a fresh state-of-the arte about biological hypercomputation and the weave of information processing in nature.

Throughout this chapter the basic argument is that life can be seen as a large cooperative game by which the increasing complexity of life lies on the very increasing processes of information processing.

## **2. A serious problem: what is information (one more time!)**

In order to explain nature band the universe, the eighteenth century invented or created a physical concept, namely mas or matter. Thanks to it, the entire universe could be explained in terms of three basic but elegant laws. This was I. Newton's achievement. Anything that is or happens is the outcome of what happens to mass, thus: relationships of action-and-reaction, the relationship between two bodies based on a force equal in magnitude and opposite in direction with each other, and the heaviness or mass of a body so that the heavier body will always attract the lighter body to itself, i.e. gravity.

In the framework of the study of heat a brand new science originating both in physics and chemistry arose in the nineteenth century that introduced a quite different new concept to understand and explain nature and reality, namely energy. Whereas mass or matter is a univocal concept, energy shows a manifold of ways of existence: caloric energy, kinetic energy, potential energy, and others. The contributions of scientists such as Fourier, Carnot, Boltzmann, Lord Kelvin, and others, had as a result the identification of three laws of thermodynamics. Those three laws were sufficient to explain not only the dynamics of the universe and nature, but also the arrow of time and the challenges it arose.

Now, the twentieth century discovered still another physical concept highly more complex and different form the two original ones. This was the concept of information, originally brought out by Shannon and Weaver, which, however, was to know still further developments manly due to the research on cryptography, quantum theory, quantum computation and quantum science, mainly. The crux here is that information is a physical and yet nontangential, non-material concept—in contrast with mass and energy [1].

Now, the relation among the three physical concepts can be adequately identified as follows:

$$\text{Matter } \mathfrak{e} \text{ Energy } \mathfrak{e} \text{ Information} \tag{1}$$

Meaning that energy explains better and deeper what matter is supposed to explain, and that information explains still much better and deeper what energy was supposed to explain. Thus, the story from classical mechanics to thermodynamics to information theory is the very story by which a higher and more achieved understanding of reality is accomplished. This, however should not imply by any means a linear and necessarily accumulative process, but a story of improving, gaining more knowledge and wisdom about the universe, nature, and society. The story of knowledge can very appropriately be grasped in terms of bifurcations [2].

freedom. This view, however, should not be understood hierarchically. Rather, the architecture is to be seen as a fractal organization. As the growing architecture of life increases, an amplification of previous lower layers is produced, that does not obliterate or supersedes the

To be sure, the process by which the living systems gain degrees of freedom coincides with the entire weave of networks, fractal architectures, and interdependencies among layers, planes, and contexts. As a result, life and nature can be seen as a fantastic interweaving of

A number of works have been done that permit the approach of this text. The various authors, takes, and approaches will be discussed along the text, which brings a fresh state-of-the arte about biological hypercomputation and the weave of information processing in nature.

Throughout this chapter the basic argument is that life can be seen as a large cooperative game by which the increasing complexity of life lies on the very increasing processes of infor-

In order to explain nature band the universe, the eighteenth century invented or created a physical concept, namely mas or matter. Thanks to it, the entire universe could be explained in terms of three basic but elegant laws. This was I. Newton's achievement. Anything that is or happens is the outcome of what happens to mass, thus: relationships of action-and-reaction, the relationship between two bodies based on a force equal in magnitude and opposite in direction with each other, and the heaviness or mass of a body so that the heavier body will

In the framework of the study of heat a brand new science originating both in physics and chemistry arose in the nineteenth century that introduced a quite different new concept to understand and explain nature and reality, namely energy. Whereas mass or matter is a univocal concept, energy shows a manifold of ways of existence: caloric energy, kinetic energy, potential energy, and others. The contributions of scientists such as Fourier, Carnot, Boltzmann, Lord Kelvin, and others, had as a result the identification of three laws of thermodynamics. Those three laws were sufficient to explain not only the dynamics of the universe

Now, the twentieth century discovered still another physical concept highly more complex and different form the two original ones. This was the concept of information, originally brought out by Shannon and Weaver, which, however, was to know still further developments manly due to the research on cryptography, quantum theory, quantum computation and quantum science, mainly. The crux here is that information is a physical and yet non-

Now, the relation among the three physical concepts can be adequately identified as follows:

**2. A serious problem: what is information (one more time!)**

always attract the lighter body to itself, i.e. gravity.

and nature, but also the arrow of time and the challenges it arose.

tangential, non-material concept—in contrast with mass and energy [1].

lower levels.

mation processing.

times, layers, and interdependencies.

84 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

The amount of information of a system depends on how probable an event is. In other words, the higher the probability that an event takes place, the lower the amount—and the quality—of the system. And vice versa: the lower the probability that an event happens, the higher—and better—the information of that event. Briefly said, the information is inversely proportional to the probability—of an event. This means that information is a measure of how surprising something is [3].

As it can be seen, the characterization I have provided about information does not necessarily coincide with the standard view—say, Shannon- that identifies information and entropy. Rather, entropy can be adequately grasped as the quantification of randomness—of a system.

Therefore, it can be easily stated that the amount of information in the universe as a whole can only increase, which means that what was considered as an expanding universe in terms of the transition from energy to matter is really an expansion of information [4]. This coincides with Formula (1) presented above. Information is the concept or reality that encompasses and explains energy and matter (mass). Whereas matter presents a view of the world rather disconnected (indeed, it is the laws that gather and unify matter, according to Newton), and energy unfolds and transforms as a set of expressions and processes explained in terms of Fourier's and Boltzmann's laws, information set out a convergent, yet in-process, reality that allows for evolution [5]. Formula (2) expresses the unity of matter, energy, and information, showing hence that there are not three things but only one—read thereafter in terms of information, according to Formula (1).

$$\text{Matter} - \text{Energy} - \text{Information} \tag{2}$$

Formula (2) serves as a synthesis that claims that matter, energy, and information are one and the same thing that were historically discovered or brought out by stages. That story is exactly the story that leads from classical mechanics to thermodynamics to information theory.

Now, summarized, information theory is cryptography, quantum information, entanglement, and teleportation [6]. These are three different faces of one and the same development—an achievement that crosses the most conspicuous technologies nowadays, as well as spearhead science in any domain you wish. By and large information pervades both science and culture, currently.

A short survey to the history of computation may be very helpful here. Indeed, information processing encounters five main levels of development, thus:

of entities but of information [7]. Exactly in this sense, too, teleportation is not so much the transportation of a physical entity, what of what makes that entity as such, namely information. It is information, indeed, what is teleported, not the thing as such, in its materiality.

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 87

Rightly understood, we can safely say that there is information processing in nature. Briefly

Living beings do not just read the environment; moreover, they accordingly unceasingly create brand new information that was non-existing before. In other words, living beings read the environment but they also, at the same time, write on the environment they are reading

Whereas any Turing Machine (TM) processes information top-down, sequentially, linearly, and mechanistically, according to the prevailing Von Neumann architecture of TM-computers, living beings, to say the least, process information in-non classical way if the framework is the

This text argues that the processing of information among living beings consists not only and not so much—in rightly reading and interpreting the environment and the surroundings, but also—and mainly—in introducing or creating brand new information into the world,

Thus, over against a popular understanding of the issue, information processing does not have anything to do with things such as "following a thread," "analyzing," nor even "understanding." Processing information can rather be grasped in biological or medical terms as metabolizing, i.e., changing one thing—say A—into another, B, for instance -. Good metabolizing, hence, transforms a nice meal into a poem or a scientific paper, or also a nice rest in the

For the living beings, processing information is a matter of life or death—in that a bad information processing may entail danger, peril or death. Biologically or culturally speaking, information processing might entail identifying a good male or female a god territory, the presence of a friend or an enemy, who truly loves you and who just pretends to love you, which food is healthy and which poisonous, for instance. Living beings that process information rightly may encounter better circumstances for adaptation and learning, and be literally the fittest. Evolution and adaptation are after all, it appears, a matter of good information processing.

Yet, life is not just an emergent property of the universe. A number of authors have claimed that the universe herself is alive and conscious [8–10]. The claim is hence about consciousness not as an epiphenomenon of the universe, and life as an emergent stance. In this take, life is an essential feature of the universe that can be traced back to the very processing of information, precisely. For the sake of brevity I shall put such a claim here into brackets and focus on the

**3. Living beings process information: a complexity understanding**

Matter is information, which brings us back to Formulas (1) and (2).

said, nature can be understood in terms of information.

on—if the analogy is permitted here.

evening into insight and strength, for example.

way in which living systems do process information.

Turing-Church thesis.

correspondingly.

It is clear that throughout the four mentioned layers, the very technological capabilities are developed and enhanced, for example via the so-called Moore law (after G. Moore, then president of Intel), that deals with the speed of change of the processors in computers. Undoubtedly, information processing has been changing the very understanding of computing and information, as well as it has been the subject—ultimately of software engineering as well as from hardware engineering—the two basic domains of computation, roughly said.

From the standpoint of quantum theory, information sets out the ground to unify epistemology and ontology. The link that allows such a connection is the wave function. Quantum science does not allow for the distinction, and even less the hierarchy, between ontology and epistemology, on either side. Instead, physics is about what we can say about nature—not about what nature is, any longer.

To be sure, the concept-behavior that allows for the unity of ontology and epistemology is the concept of entanglement, a concept firstly introduced by Schrödinger (*Vershränkung*), but truly implemented by J. Bell. When entangled, the unity of two or more particles becomes more important than the particles individually considered. Entanglement is not really entanglement of entities but of information [7]. Exactly in this sense, too, teleportation is not so much the transportation of a physical entity, what of what makes that entity as such, namely information. It is information, indeed, what is teleported, not the thing as such, in its materiality. Matter is information, which brings us back to Formulas (1) and (2).

Rightly understood, we can safely say that there is information processing in nature. Briefly said, nature can be understood in terms of information.

## **3. Living beings process information: a complexity understanding**

A short survey to the history of computation may be very helpful here. Indeed, information

• Linear, sequential, top-down information processing. Amply, this is the largely predominant paradigm in computation and computing science. Ever since the first computers, Eniac I, Eniac II, etc., on until now, computers and computations are defined by the Von Neumann architecture, and the Church-Turing thesis. A programmer programs a program and the computer follows the indications and directions. Design, parametrization, control are the key operations and ways of working and dealing with computers and computation. • Dynamic and evolving information processing. Genetic algorithms, time series, and the first steps in artificial intelligence (AI) such as the works by Ray, Conway, and neural networks, can be said to define the second layer of information processing. The shift toward bottom-up concerns and interests is produced that largely enhance computation as a dynamic process. Not eventually modeling and simulation come along with literal explosion

• Biologically inspired information processing. More recently bio-inspired computation takes as model or guide the behavior of living systems—say, DNA, proteins, ant colonies, etc.—and develops programs that come closer to the way living beings behave. This kind of information processing has shed so far brand new fresh lights on to the very way in which computing science had being previously conceived. Thus, computation and biology at large come closer and learn from each other making computation much more flexible

• Biological hypercomputation. On the basis of the third layer, trying to understand the way in which living beings process information clearly entails a distance from classical computation, no matter what. Such a computation has been named biological hypercomputation.

It is clear that throughout the four mentioned layers, the very technological capabilities are developed and enhanced, for example via the so-called Moore law (after G. Moore, then president of Intel), that deals with the speed of change of the processors in computers. Undoubtedly, information processing has been changing the very understanding of computing and information, as well as it has been the subject—ultimately of software engineering as well as from hardware engineering—the two basic domains of computation, roughly said.

From the standpoint of quantum theory, information sets out the ground to unify epistemology and ontology. The link that allows such a connection is the wave function. Quantum science does not allow for the distinction, and even less the hierarchy, between ontology and epistemology, on either side. Instead, physics is about what we can say about nature—not

To be sure, the concept-behavior that allows for the unity of ontology and epistemology is the concept of entanglement, a concept firstly introduced by Schrödinger (*Vershränkung*), but truly implemented by J. Bell. When entangled, the unity of two or more particles becomes more important than the particles individually considered. Entanglement is not really entanglement

processing encounters five main levels of development, thus:

86 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

of programming languages aimed at a variety of goals and abilities.

and robust at the same time. Adaptive computation comes to the fore.

This is the core of this text.

about what nature is, any longer.

Living beings do not just read the environment; moreover, they accordingly unceasingly create brand new information that was non-existing before. In other words, living beings read the environment but they also, at the same time, write on the environment they are reading on—if the analogy is permitted here.

Whereas any Turing Machine (TM) processes information top-down, sequentially, linearly, and mechanistically, according to the prevailing Von Neumann architecture of TM-computers, living beings, to say the least, process information in-non classical way if the framework is the Turing-Church thesis.

This text argues that the processing of information among living beings consists not only and not so much—in rightly reading and interpreting the environment and the surroundings, but also—and mainly—in introducing or creating brand new information into the world, correspondingly.

Thus, over against a popular understanding of the issue, information processing does not have anything to do with things such as "following a thread," "analyzing," nor even "understanding." Processing information can rather be grasped in biological or medical terms as metabolizing, i.e., changing one thing—say A—into another, B, for instance -. Good metabolizing, hence, transforms a nice meal into a poem or a scientific paper, or also a nice rest in the evening into insight and strength, for example.

For the living beings, processing information is a matter of life or death—in that a bad information processing may entail danger, peril or death. Biologically or culturally speaking, information processing might entail identifying a good male or female a god territory, the presence of a friend or an enemy, who truly loves you and who just pretends to love you, which food is healthy and which poisonous, for instance. Living beings that process information rightly may encounter better circumstances for adaptation and learning, and be literally the fittest. Evolution and adaptation are after all, it appears, a matter of good information processing.

Yet, life is not just an emergent property of the universe. A number of authors have claimed that the universe herself is alive and conscious [8–10]. The claim is hence about consciousness not as an epiphenomenon of the universe, and life as an emergent stance. In this take, life is an essential feature of the universe that can be traced back to the very processing of information, precisely. For the sake of brevity I shall put such a claim here into brackets and focus on the way in which living systems do process information.

Whereas at the same time living beings—from bacteria to cells, from organisms to biomes, for example—continuously read the environment and the surroundings they create brand new information that is brought into the world in a variety of forms, thus: as acts and actions, as behaviors, as creation in same cases of tools and rods, as the use of tools as devices aimed for a certain goal, definitely as language and communication, and as forms of organization, whether individual or collective [11]. It is the very creation of new information into the world, which is usually grasped as adaptation. Adaptation is the biological concept that originally is the outcome of information processing.

or even symbolic epigenetics, becomes central here, for it most adequately expresses the idea that very much as there is no difference between software and hardware, in the same tenure, there is no difference between nature and culture. They are closely intertwined and cannot be

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 89

As a consequence, computing (= BH) is for life one and the same thing with evolution. Evolving and computing are two faces of one and the same token. Evolution is the process through

Information processing among living beings takes place as an unceasing creation of information that is introduced into the world. Thus, information processing is not just reading the environment and adequately interpreting it, but, moreover, correspondingly, bringing new information into the world in the form of actions, processes, behaviors, or achievements.

There is no information before the information processing by living beings, very much as there is no information after the information processing by living beings. Information exists in so far as it is processed, that is, both created and changed. In other words, there is no reality previous to the processing of information, but neither is there any reality "outside" the information processing. Yet, the universe becomes increasingly complex precisely thanks to the unceasing processing of information, i.e. the reduction of entropy in the universe. Life is

As a consequence, the universe is alive it appears. A number of interpretations in this sense can be mentioned that challenge the traditional view of the universe that goes as follows:

PhysicsInorganic ChemistryOrganic ChemistryBiologyCulture (3)

I argue that Formula (3) is erroneous because it is set on the physical predominance of physics over biochemistry or biology, for example. Formula (3) is really a translation of Formula (1), and thus, it sets out the ground for a reductionist view of nature and the universe. A clear

The increasing complexity of reality and the universe is one and the same process as the increasing complexity in the information processes [13, 14]. We have been gaining an enormous field in understanding what life does and how it is made possible. Thus, for instance, regarding the realm of plants, Refs. [15, 16] have shed solid lights about how they think, literally. Down the scale, [17] did steadily study the way in which information processing takes place among bacteria. Furthermore, information processing has been studied additionally among social insect networks [18]. The range of studies and the cope is always wider and deeper, encompassing animals at large [19], until of course we reach the level of human

that system that creates order and exhibits the best form of order possible.

reductionist approach serves as ground for that formula.

beings, and computers and computation.

divided or split.

which living beings gain degrees of freedom.

**4. Living beings gain degrees of freedom**

As it can be easily seen, living beings compute. Moreover, life without computation is inconceivable.

Computation is therefore a concept that truly means interaction—with the environment, and with other living beings. Natural computation is much more than a metaphor, but the distinguishing feature of living beings. Life is a large weave of computation that takes place in a manifold of ways simultaneously, as follows:

There is classical computation in nature in either form of a Turing Machine. At the same time there is parallel computation, and multilevel computation, non-local computation as well as distributed computation, fuzzy and random computation very much as also quantum computation and emergent computation, not to mention interactive computation. A typology of computation and computational models can be seen in [12]. A metaphor can be introduced here, namely a variety of computations corresponds to the diversity of life and living beings. It appears that ecology and biology go hand in hand with information theory and computing science, even though the two latter are more recent that the two before [13].

To be sure, biological hypercomputation is the way in which living beings process information—is anything but machines. However, there are a number of explanations within the health sciences and the living sciences that still operate mechanically. Thus, for instance, the functioning of the brain many times is explained in terms of "on" and "off" switches or operations, and the very functioning of the heart is usually conceived as a pumping machine, period. Many other cases can be introduced here. The crux is that those explanations still owe a big deal to the past and remain very short vis-à-vis more contemporary and spearhead explanations. Computing sciences is such one of those frontline explanations, under the proviso that it be not a Turing Machine one, in any concern.

Biological hypercomputation (BH) has been introduced [15] to mean that life and the living processes cannot be understood from lower stances: in other words, life is to be explained as it happens: life is an uncompressible "program." The most crucial and basic problems of living beings are certainly not tackled in terms of mathematical functions, whatsoever. Instead, the problems living beings faced are normally solved in terms of non-classical logics, a field that has not deserved as much attention as its own value.

More fundamentally still, computationally speaking, the difference between software and hardware is irrelevant. More exactly, it does not exist. The phenotype expresses the genotype according to evolution and the environment. Epigenetics, for example behavioral epigenetics or even symbolic epigenetics, becomes central here, for it most adequately expresses the idea that very much as there is no difference between software and hardware, in the same tenure, there is no difference between nature and culture. They are closely intertwined and cannot be divided or split.

As a consequence, computing (= BH) is for life one and the same thing with evolution. Evolving and computing are two faces of one and the same token. Evolution is the process through which living beings gain degrees of freedom.

## **4. Living beings gain degrees of freedom**

Whereas at the same time living beings—from bacteria to cells, from organisms to biomes, for example—continuously read the environment and the surroundings they create brand new information that is brought into the world in a variety of forms, thus: as acts and actions, as behaviors, as creation in same cases of tools and rods, as the use of tools as devices aimed for a certain goal, definitely as language and communication, and as forms of organization, whether individual or collective [11]. It is the very creation of new information into the world, which is usually grasped as adaptation. Adaptation is the biological concept that originally is

As it can be easily seen, living beings compute. Moreover, life without computation is

Computation is therefore a concept that truly means interaction—with the environment, and with other living beings. Natural computation is much more than a metaphor, but the distinguishing feature of living beings. Life is a large weave of computation that takes place in a

There is classical computation in nature in either form of a Turing Machine. At the same time there is parallel computation, and multilevel computation, non-local computation as well as distributed computation, fuzzy and random computation very much as also quantum computation and emergent computation, not to mention interactive computation. A typology of computation and computational models can be seen in [12]. A metaphor can be introduced here, namely a variety of computations corresponds to the diversity of life and living beings. It appears that ecology and biology go hand in hand with information theory and computing

To be sure, biological hypercomputation is the way in which living beings process information—is anything but machines. However, there are a number of explanations within the health sciences and the living sciences that still operate mechanically. Thus, for instance, the functioning of the brain many times is explained in terms of "on" and "off" switches or operations, and the very functioning of the heart is usually conceived as a pumping machine, period. Many other cases can be introduced here. The crux is that those explanations still owe a big deal to the past and remain very short vis-à-vis more contemporary and spearhead explanations. Computing sciences is such one of those frontline explanations, under the pro-

Biological hypercomputation (BH) has been introduced [15] to mean that life and the living processes cannot be understood from lower stances: in other words, life is to be explained as it happens: life is an uncompressible "program." The most crucial and basic problems of living beings are certainly not tackled in terms of mathematical functions, whatsoever. Instead, the problems living beings faced are normally solved in terms of non-classical logics, a field that

More fundamentally still, computationally speaking, the difference between software and hardware is irrelevant. More exactly, it does not exist. The phenotype expresses the genotype according to evolution and the environment. Epigenetics, for example behavioral epigenetics

science, even though the two latter are more recent that the two before [13].

viso that it be not a Turing Machine one, in any concern.

has not deserved as much attention as its own value.

the outcome of information processing.

88 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

manifold of ways simultaneously, as follows:

inconceivable.

Information processing among living beings takes place as an unceasing creation of information that is introduced into the world. Thus, information processing is not just reading the environment and adequately interpreting it, but, moreover, correspondingly, bringing new information into the world in the form of actions, processes, behaviors, or achievements.

There is no information before the information processing by living beings, very much as there is no information after the information processing by living beings. Information exists in so far as it is processed, that is, both created and changed. In other words, there is no reality previous to the processing of information, but neither is there any reality "outside" the information processing. Yet, the universe becomes increasingly complex precisely thanks to the unceasing processing of information, i.e. the reduction of entropy in the universe. Life is that system that creates order and exhibits the best form of order possible.

As a consequence, the universe is alive it appears. A number of interpretations in this sense can be mentioned that challenge the traditional view of the universe that goes as follows:

PhysicsInorganic ChemistryOrganic ChemistryBiologyCulture (3)

I argue that Formula (3) is erroneous because it is set on the physical predominance of physics over biochemistry or biology, for example. Formula (3) is really a translation of Formula (1), and thus, it sets out the ground for a reductionist view of nature and the universe. A clear reductionist approach serves as ground for that formula.

The increasing complexity of reality and the universe is one and the same process as the increasing complexity in the information processes [13, 14]. We have been gaining an enormous field in understanding what life does and how it is made possible. Thus, for instance, regarding the realm of plants, Refs. [15, 16] have shed solid lights about how they think, literally. Down the scale, [17] did steadily study the way in which information processing takes place among bacteria. Furthermore, information processing has been studied additionally among social insect networks [18]. The range of studies and the cope is always wider and deeper, encompassing animals at large [19], until of course we reach the level of human beings, and computers and computation.

The concept "degrees of freedom" can be understood from a wide take that says that it consists in the number of parameters a system exhibits or has that may vary independently (Wikipedia) until the calculation of the various scenarios in a system or the number of dimensions that are needed to determine a full vector. In complexity science or theory the concept is pivotal and allows for crossing different approaches, sciences, discipline, and interpretations aiming at making explicit what complexity is all about.

We can safely say that the higher the degrees of freedom a system has the more complex it is. And vice versa, the lower the degrees of freedom, the less complex it is. Formula (4) synthetizes this:

$$\text{\textquotedblleft Degrees of Freedom \textquotedblright} \text{Complexity} \tag{4}$$

**5. The weave of life**

versa.

on or very near the equator.

that *telos* are necessary.

story of learning, adaptation and coevolution.

Information is a continuous process that emerges as a result of intricate non-linear interactions of living beings among themselves and with their environment that both creates nature and the world as it happens, and at the same time changes it unceasingly. This is exactly the

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 91

As it is well known, living beings evolve in rugged adaptive landscapes—a concept originally introduced by Ch. Darwin in 1859, originally named as fitness landscape. Life is a wonderful complex weave that has no center, but many bubs, clusters and nodes, continuously changing on the ever-changing environment on earth. Only that the timescales in geology are vastly deeper than those of the living beings, not to mention human beings. Indeed, whereas the basic time-scale of life is the decade or even the month for some species, the basic time-scale in geology is 1000.000 years. Against all odds, this is the ultimate time-scale of life on earth.

Life is a large and robust weave of cooperation, commensalism, and mutualism, rather than competition and predation. [20] has made on this subject a great contribution, namely the importance of symbiogenesis: a large network (instead than a chain) of mutual interdependency among living beings so that each organism and species benefits from others, and vice

The story of life is a story of increasing complexity, i.e. biodiversity. Such diversity is said to be at three levels, thus: genetic, natural, and cultural. The countries that have the three kinds of biodiversity are called "megadiverse" and they are 17 countries, to-date—all of them place

The story of life pivots around six fantastic moments of increasing diversity followed by six massive extinctions. At each stage, life has made of herself a wonderful asset of possibilities, forms, shapes, structures, behaviors, and characteristics that correspond with dynamics through which, as a whole, amazing degrees of freedom are been attained or reached. And yet, that story is non-teleological. In other words, life has no ends or goals, and evolution has no purpose whatsoever. Such was indeed the scandal that Darwin's *The origins of species by means of natural selection* (1859) meant particularly vis-à-vis the cornerstone of the western civilization, namely teleology [21]. The western world needs believe that there are *telos*, and

To the discomfort of those who believe in goals end, whether their own or imposed or suggested, the theory of evolution—namely the best theory ever developed to think about change and transformation—introduces the idea of the absence of goals or ends (*telos*). Well, the theory of symbiogenesis comes to complete, so to speak, the theory of evolution by showing that the various stages of complexity of life, firstly, and secondly that the whole weave of life naturally tends toward increasing complexity and hence to cooperation and interdependency.

As a consequence, living beings are symbiont: not only it is a cooperative interaction but also a prolonged one among different species and/or organisms. Mutualism and not competition

is the rule in nature, it appears. Nature is the realm of freedom par excellence.

Now, living beings are what they do. And the best they do is information processing, exactly. The better a living system processes information, the fittest it is. As a consequence, the more degrees of freedom it gains. Hence, living is a matter of gaining degrees of freedom. Death, it seems, is the complete loss of degrees of freedom. All possibilities are then closed.

Therefore the claim can be made that information plays an "active" role in the living systems, and a "passive" role in classical physics. Analogously as it happens with any Turing Machine, namely information is something that "happens" in the processor that is conceived as a black box. The audience or the researcher knows what enters (input) and knows what comes out (output), but ignores what and how happens in the black box. Classical computation seems to reduce the degrees of freedom by accepting or assessing the processing of information as a "black box." On the contrary, (BH) is the process by which new degrees of freedom are gained in so far as new adaptations, networks and behaviors are made possible, for instance?

Information processing does not stand as a condition for evolution and adaptation. On the contrary, it is the very process of evolution both of the living beings and the environment. Coevolution is the name that best suits both stances in their entanglement. Living is nothing else that coping with the information of the surroundings and accordingly changing that information in terms of brand new information brought into the world.

The better the living system processes information, the better it copes with challenges, solves problems, adapts to a continuously changing environment. Life ends when it "overdoses" of information and the living being is unable to process it. A saturation point is reached and the system can no longer process any further information [20].

Summarizing, life consists in the process by which information processing is carried out and hence unceasingly new degrees of freedom are reached. The system can be said to be "young." Contrarily, death happens when the processing of information is not possible any longer—it becomes slow, saturated, new information cannot be processed accordingly, and the "screen" slows down in bringing out the information required.

Life is a matter of "controlling" the saturation of information, and thereafter, of information processing. Software and hardware are one and the same thing.

## **5. The weave of life**

The concept "degrees of freedom" can be understood from a wide take that says that it consists in the number of parameters a system exhibits or has that may vary independently (Wikipedia) until the calculation of the various scenarios in a system or the number of dimensions that are needed to determine a full vector. In complexity science or theory the concept is pivotal and allows for crossing different approaches, sciences, discipline, and interpretations

We can safely say that the higher the degrees of freedom a system has the more complex it is. And vice versa, the lower the degrees of freedom, the less complex it is. Formula (4) synthe-

> Degrees of Freedom > Complexity (4)

Now, living beings are what they do. And the best they do is information processing, exactly. The better a living system processes information, the fittest it is. As a consequence, the more degrees of freedom it gains. Hence, living is a matter of gaining degrees of freedom. Death, it

Therefore the claim can be made that information plays an "active" role in the living systems, and a "passive" role in classical physics. Analogously as it happens with any Turing Machine, namely information is something that "happens" in the processor that is conceived as a black box. The audience or the researcher knows what enters (input) and knows what comes out (output), but ignores what and how happens in the black box. Classical computation seems to reduce the degrees of freedom by accepting or assessing the processing of information as a "black box." On the contrary, (BH) is the process by which new degrees of freedom are gained

seems, is the complete loss of degrees of freedom. All possibilities are then closed.

in so far as new adaptations, networks and behaviors are made possible, for instance?

information in terms of brand new information brought into the world.

system can no longer process any further information [20].

the "screen" slows down in bringing out the information required.

processing. Software and hardware are one and the same thing.

Information processing does not stand as a condition for evolution and adaptation. On the contrary, it is the very process of evolution both of the living beings and the environment. Coevolution is the name that best suits both stances in their entanglement. Living is nothing else that coping with the information of the surroundings and accordingly changing that

The better the living system processes information, the better it copes with challenges, solves problems, adapts to a continuously changing environment. Life ends when it "overdoses" of information and the living being is unable to process it. A saturation point is reached and the

Summarizing, life consists in the process by which information processing is carried out and hence unceasingly new degrees of freedom are reached. The system can be said to be "young." Contrarily, death happens when the processing of information is not possible any longer—it becomes slow, saturated, new information cannot be processed accordingly, and

Life is a matter of "controlling" the saturation of information, and thereafter, of information

aiming at making explicit what complexity is all about.

90 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

tizes this:

Information is a continuous process that emerges as a result of intricate non-linear interactions of living beings among themselves and with their environment that both creates nature and the world as it happens, and at the same time changes it unceasingly. This is exactly the story of learning, adaptation and coevolution.

As it is well known, living beings evolve in rugged adaptive landscapes—a concept originally introduced by Ch. Darwin in 1859, originally named as fitness landscape. Life is a wonderful complex weave that has no center, but many bubs, clusters and nodes, continuously changing on the ever-changing environment on earth. Only that the timescales in geology are vastly deeper than those of the living beings, not to mention human beings. Indeed, whereas the basic time-scale of life is the decade or even the month for some species, the basic time-scale in geology is 1000.000 years. Against all odds, this is the ultimate time-scale of life on earth.

Life is a large and robust weave of cooperation, commensalism, and mutualism, rather than competition and predation. [20] has made on this subject a great contribution, namely the importance of symbiogenesis: a large network (instead than a chain) of mutual interdependency among living beings so that each organism and species benefits from others, and vice versa.

The story of life is a story of increasing complexity, i.e. biodiversity. Such diversity is said to be at three levels, thus: genetic, natural, and cultural. The countries that have the three kinds of biodiversity are called "megadiverse" and they are 17 countries, to-date—all of them place on or very near the equator.

The story of life pivots around six fantastic moments of increasing diversity followed by six massive extinctions. At each stage, life has made of herself a wonderful asset of possibilities, forms, shapes, structures, behaviors, and characteristics that correspond with dynamics through which, as a whole, amazing degrees of freedom are been attained or reached. And yet, that story is non-teleological. In other words, life has no ends or goals, and evolution has no purpose whatsoever. Such was indeed the scandal that Darwin's *The origins of species by means of natural selection* (1859) meant particularly vis-à-vis the cornerstone of the western civilization, namely teleology [21]. The western world needs believe that there are *telos*, and that *telos* are necessary.

To the discomfort of those who believe in goals end, whether their own or imposed or suggested, the theory of evolution—namely the best theory ever developed to think about change and transformation—introduces the idea of the absence of goals or ends (*telos*). Well, the theory of symbiogenesis comes to complete, so to speak, the theory of evolution by showing that the various stages of complexity of life, firstly, and secondly that the whole weave of life naturally tends toward increasing complexity and hence to cooperation and interdependency.

As a consequence, living beings are symbiont: not only it is a cooperative interaction but also a prolonged one among different species and/or organisms. Mutualism and not competition is the rule in nature, it appears. Nature is the realm of freedom par excellence.

#### **6. Conclusions**

We have three concepts: matter, energy and information. However, a right understanding of them brings to the fore the fact that they are not three, but one and the same concept: matter-energy-information. Moreover, information can be rightly understood as fundamental (= grounding) energy and mass.

What happens to matter is truly a matter of energy and relations among various types of energy. And still the very processes, dynamics and configurations of energy are dynamics in and of information. Formula (4) expresses this understanding.

$$\text{Information} \le \text{Energy} \le \text{Matter (Mass)}\tag{5}$$

[4] Roederer JG. Information and Its Role in Nature. Springer Verlag; 2005

University Press; 2001

Vintage Books; 2008

pe=pdf

2010

2012;**7**(7):e40337

Vintage Books; 2003

Farrar, Straus and Giroux; 2010

Theory. Springer Verlag; 1990

[5] Chaisson E. Cosmic Evolution. The Rise of Complexity in Nature. Harvard: Harvard

Biological Hypercomputation and Degrees of Freedom http://dx.doi.org/10.5772/intechopen.73179 93

[6] Zeilinger A. Dance of the Photons. From Einstein to Quantum Teleportation. New York:

[7] Gilder L. The Age of Entanglement. When Quantum Physics was Reborn. New York:

[8] Kafatos M, Nadeau R. The Conscious Universe. Part and Whole in Modern Physical

[9] Gribbin J. In the Beginning: The Birth of the Living Universe. Little Brown & Co.; 1995

[10] Kauffman S. Humanity in a Creative Universe. Oxford: Oxford University Press; 2016

Ciencias Exactas, Físicas y Naturales; 2017. pp. 109-124

Cambridge: Cambridge University Press; 2013

New York Academy of Sciences. Oct 2009;**1178**:78-90

Animal Studies. New York: Columbia University Press; 2012

Intelligence. Island Press; 2015

[11] Maldonado CE. Hipercomputación biológica y comunicación entre los seres vivos. In: Arboleda LC, editor. Un Festschrift para José Luis Villaveces. Bogotá: Academia de

[12] Burgin M, Dodig-Crnkovic G. Typologies of Computation and Computational Models; 2013. http://citeseerx.ist.psu.edu/viewdoc/download?doi=<IP_ADDRESS>6.5721&rep=rep1&ty

[13] Linewaever CH, Davies PCW, Ruse M, editors. Complexity and the Arrow of Time.

[14] Barrow JD, Davies PCW, Harper CL Jr, editors. Science and Ultimate Reality. Quantum Theory, Cosmology and Complexity. Cambridge: Cambridge University Press; 2005

[15] Maldonado CE, Gómez-Cruz N. Biological hypercomputation: A new research problem

[16] Mancuso S, Viola A. Brilliant Green. The Surprising History and Science of Plant

[17] Baluska F. Signaling in Plants (Signaling and Communication in Plants). Springer Verlag;

[18] Ben-Jacob E. Learning from bacteria about natural information processing. Annals of the

[19] Waters JS, Fewell JH. Information processing in social insect networks. PLoS One.

[20] Gross A, Vallely A, editors. Animals and the Human Imagination. A Companion to

[21] Margulis L, Sagan D. Acquiring Genomes. A Theory of the Origins of Species. New York:

in complexity theory. Complexity. 2015;**20**(4):8-18. ISSN 1099-0526

Formula (5) simply means that matter explains less (and worst) what energy does explain, and furthermore, information allows for a deeper and better understanding of what energy means to explain.

Now, when looked under the light of life and how life is possible, the three concepts are transformed into one process, namely biological hypercomputation. In the processing of information among the living beings matter, energy and information become one single unity: life. Thus, life allows for the overcoming three different layers of reality (physics, thermodynamics, and information); such overcoming is also the non-differentiation between software and hardware. Processing information for life consists in gaining degrees of freedom.

Living beings process information much more than (just) matter and energy, which they definitely do. Evolution can be seen as the processing of information processing by living beings through which the world and nature are complexified. Information allows for the rejection of a reductionist approach of life and nature.

#### **Author details**

Carlos Eduardo Maldonado

Address all correspondence to<EMAIL_ADDRESS>
School of Medicine, Universidad El Bosque, Bogotá, Colombia

#### **References**

[4] Roederer JG. Information and Its Role in Nature. Springer Verlag; 2005

**6. Conclusions**

means to explain.

**Author details**

**References**

Carlos Eduardo Maldonado

University Press; 2010

(= grounding) energy and mass.

a reductionist approach of life and nature.

We have three concepts: matter, energy and information. However, a right understanding of them brings to the fore the fact that they are not three, but one and the same concept: matter-energy-information. Moreover, information can be rightly understood as fundamental

What happens to matter is truly a matter of energy and relations among various types of energy. And still the very processes, dynamics and configurations of energy are dynamics in

Information ≤ Energy ≤ Matter (Mass) (5)

Formula (5) simply means that matter explains less (and worst) what energy does explain, and furthermore, information allows for a deeper and better understanding of what energy

Now, when looked under the light of life and how life is possible, the three concepts are transformed into one process, namely biological hypercomputation. In the processing of information among the living beings matter, energy and information become one single unity: life. Thus, life allows for the overcoming three different layers of reality (physics, thermodynamics, and information); such overcoming is also the non-differentiation between software and

Living beings process information much more than (just) matter and energy, which they definitely do. Evolution can be seen as the processing of information processing by living beings through which the world and nature are complexified. Information allows for the rejection of

[1] Gleick J. The Information: A History, a Theory, a Flood. New York: Vintage Books; 2012

[3] Vedral V. Decoding Reality. The Universe as Quantum Information. Oxford: Oxford

hardware. Processing information for life consists in gaining degrees of freedom.

Address all correspondence to<EMAIL_ADDRESS>
School of Medicine, Universidad El Bosque, Bogotá, Colombia

[2] Serres M. Éléments d'histoire des sciences. Bordas; 1993

and of information. Formula (4) expresses this understanding.

92 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

**Section 2**

**Complexity in Physical Systems**

**Complexity in Physical Systems**

**Chapter 6**

Provisional chapter

**Self-Organization, Coherence and Turbulence in Laser**

DOI: 10.5772/intechopen.71478

In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubicquintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent "internal modes" that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure,

Keywords: optical turbulence, dissipative solitons, chaos in nonlinear optical systems, generalized cubic-quintic nonlinear Ginzburg-Landau equation,

Coherent and partially coherent structures emerging in nonlinear systems far from the thermodynamic equilibrium play an important role in different research areas ranging from hydrodynamics and plasma physics to biophysics and sociology. Nontrivial dynamics of such

> © The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

Self-Organization, Coherence and Turbulence

Vladimir L. Kalashnikov and Evgeni Sorokin

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Vladimir L. Kalashnikov and Evgeni Sorokin

http://dx.doi.org/10.5772/intechopen.71478

namely, a dissipative Raman soliton.

dissipative Raman soliton

1. Introduction

**Optics**

in Laser Optics

Abstract

Provisional chapter

#### **Self-Organization, Coherence and Turbulence in Laser Optics** Self-Organization, Coherence and Turbulence

DOI: 10.5772/intechopen.71478

in Laser Optics

Vladimir L. Kalashnikov and Evgeni Sorokin

Additional information is available at the end of the chapter Vladimir L. Kalashnikov and Evgeni Sorokin

http://dx.doi.org/10.5772/intechopen.71478 Additional information is available at the end of the chapter

#### Abstract

In the last decades, rapid progress in modern nonlinear science was marked by the development of the concept of dissipative soliton (DS). This concept is highly useful in many different fields of science ranging from field theory, optics, and condensed matter physics to biology, medicine, and even sociology. This chapter aims to present a DS appearance from random fluctuations, development, and growth, the formation of the nontrivial internal structure of mature DS and its breakup, in other words, a full life cycle of DS as a self-organized object. Our extensive numerical simulations of the generalized cubicquintic nonlinear Ginzburg-Landau equation, which models, in particular, dynamics of mode-locked fiber lasers, demonstrate a close analogy between the properties of DS and the general properties of turbulent and chaotic systems. In particular, we show a disintegration of DS into a noncoherent (or partially coherent) multisoliton complex. Thus, a DS can be interpreted as a complex of nonlinearly coupled coherent "internal modes" that allows developing the kinetic and thermodynamic theory of the nonequilibrious dissipative phenomena. Also, we demonstrate an improvement of DS integrity and, as a result, its disintegration suppression due to noninstantaneous nonlinearity caused by the stimulated Raman scattering. This effect leads to an appearance of a new coherent structure, namely, a dissipative Raman soliton.

Keywords: optical turbulence, dissipative solitons, chaos in nonlinear optical systems, generalized cubic-quintic nonlinear Ginzburg-Landau equation, dissipative Raman soliton

#### 1. Introduction

Coherent and partially coherent structures emerging in nonlinear systems far from the thermodynamic equilibrium play an important role in different research areas ranging from hydrodynamics and plasma physics to biophysics and sociology. Nontrivial dynamics of such

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

structures including chaos and turbulence is a challenge for modern nonlinear science and one may assume that "the problem of turbulence is one of the central problems in theoretical physics" [1]. The reasonable approach to this issue, which can translate some contra-intuitive and obscure ideas in this area into explicit and verifiable concepts, is a realization of simpler dynamics in quite different material context. Such an approach can be named metaphorical or analog modeling [2], and a rapid progress of modern laser technology provides an ideal playground for such enterprise due to high controllability, relative simplicity, and unique potential of statistic gathering. Such progress was marked by the development of the concept of a dissipative soliton (DS) [3]. The existence of DS under nonequilibrium conditions requires a well-organized energy exchange with an environment so that this energy flow forms a nontrivial internal structure of DS, which provides the energy redistribution inside it and can distort the soliton coherence. Such a DS with nontrivial internal structure can develop in lasers, and the DS dynamics can become chaotic and turbulent [3–5]. For instance, such emergent structures can be considered as a classical analog of Bose-Einstein condensate in low dissipative limit and, contrariwise, as a primitive analog of cell in the case of extensive and wellstructured energy exchange with an environment. Formally, these inherently nonHamiltonian entities mimic some features of Hamiltonian systems that remain an obscure and insufficiently explored topic regarding the fundamental properties of coherent dissipative structures. The range of turbulence, noise, and rogue wave phenomena emulated by the optical DS is so broad that it turns them into a universal testbed for studies in the fields of nonlinear dynamical systems and nonequilibrium thermodynamics.

describing an extremely broad class of phenomena. It is possible that the most known one is the famous nonlinear Schrödinger equation (NSE) which describes an evolution of slowly varying wave in a nonlinear medium and can be considered as a "metaphoric" simulation tool

> ∂<sup>2</sup>ω ∂kj∂kl

The dimensionality of this equation is relative: the evolutional coordinate can be a time T or a propagation distance z (T\$ z), the transverse coordinate can be transverse multidimensional spatial xj (j = 1…d) one or a local time t (x\$ t, d = 1). The Fourier representations of a "field" slowly varying envelope Ψ are interchangeable between frequency and momentum domains (ω\$ k, d = 1). Eq. (1) may describe the propagation of optical pulses in a nonlinear medium (then Ψ is a complex field amplitude and |Ψ|2 is proportional to a field power), the capillary waves on a fluid surface, the Langmuir waves in plasma, or the weakly nonlinear Bose-gas in classic limit (in the last case Eq. (1) represents the famous time-dependent Gross-Pitaevskii equation [16]).

term in Eq. (1) can have the different forms; in particular, a nonlinear response can be non-

The notion of turbulence is fuzzy in some sense. Here, the turbulence will be treated as a phenomenon related to the excitation of a sufficiently large number of degrees of freedom that causes a loss of their mutual phase information [15]. As a consequence, a wave package decouples into a set of individual modes ("particles") which interaction can be described in the framework of kinetic theory as many-particle collisions in Bose-gas. In other words, as some degrees of freedom become very large for sufficiently large energies, phase information becomes irrelevant, and the waves decohere [8, 15]. Thus, a wave can be considered as a set of

Thus, we come to a "kinetic" theory of turbulence, for example, to a model of four-boson

Such an equation becomes nontrivial in a dissipative environment [17, 18]. A simple generalization of NSE (1) taking into account the dissipative effects includes a saturable gain (energy

"source") σ, dissipative nonlinearity (self-amplitude modulation, SAM) Ϝ(|Ψ|2

dissipation (spectral in the sense of dissipation in the Fourier space) P<sup>d</sup>

δ ωð Þ þ ω<sup>1</sup> � ω<sup>2</sup> � ω<sup>3</sup> d k<sup>1</sup>

nk1nk2nk<sup>3</sup> þ nknk2nk<sup>3</sup> � nknk1nk<sup>2</sup> � nknk1nk<sup>3</sup> ð Þ

Here Ψ(x, t) is a slowly varying amplitude of wave propagating in dispersive (P<sup>d</sup>

<sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> � �j j <sup>Ψ</sup> <sup>2</sup>

) and nonlinear ( <sup>∂</sup><sup>ω</sup>

decoupled "modes" nk in a spectral (or wave-number) space:

interaction described by the nonlinear Schrödinger equation:

� � !

∂<sup>2</sup>Ψ ∂xj∂xl <sup>þ</sup> <sup>i</sup> <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> !

<sup>Ψ</sup> - term; let us <sup>γ</sup> � <sup>∂</sup><sup>ω</sup>

Ψð Þk Ψ k<sup>0</sup> h i ð Þ ¼ nkδ k � k<sup>0</sup> ð Þ: (2)

! d k<sup>2</sup> ! d k<sup>3</sup> ! :

j j <sup>Ψ</sup> <sup>2</sup>

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

Ψ ¼ 0: (1)

99

j,l¼1 ∂2 ω ∂kj∂kl ∂2Ψ ∂xj∂xl -

(3)

), and spectral

Ψ ∂xj∂xl :

j,l¼<sup>1</sup> <sup>α</sup> xj; xl � � <sup>∂</sup><sup>2</sup>

<sup>∂</sup>j j <sup>Ψ</sup> 2) medium. The nonlinear

for a study of nonlinear phenomena far from equilibrium [14, 15]:

j¼1 ∂ω ∂kj ∂Ψ ∂xj � i 2 X<sup>d</sup> j,l¼1

∂Ψ <sup>∂</sup><sup>T</sup> <sup>þ</sup>X<sup>d</sup>

term; let us <sup>β</sup> � <sup>∂</sup>2<sup>ω</sup>

instantaneous.

∂kj∂kl

∂nk <sup>∂</sup><sup>t</sup> <sup>∝</sup> ð

> � δ k ! þ k<sup>1</sup> ! � k<sup>2</sup> ! � k<sup>3</sup>

In this work, we conjecture a spectacular analogy between the spectral structures of DS and strong Langmuir turbulence. Such close relation leads to chaotization of DS dynamics with the energy growth. This analogy is deepened by analysis of energy flows inside DS so that a DS can be represented as a "glass of boiling water" or, mathematically, as an ensemble of interacting quasi-particles or "nonlinear modes." The phase decoupling of these "modes" leads to turbulence or DS dissolving. Such a representation open the door for building the kinetic theory of open (dissipative) semi-coherent structures which mimics, in particular, a quantum Bose-Einstein condensate in a dissipative environment. Moreover, our preliminary investigations demonstrated a mechanism of turbulence control provided by noninstantaneous nonlinearity (stimulated Raman scattering in optical case) [6]. This phenomenon is especially interesting because an inherently noisy process (Raman scattering) suppresses a turbulence under some conditions that is the manifestation of stochastic resonance, which can be significant for a dissipation control in coherent quantum systems (particularly, a quantum computer and a quantum cryptography device).

#### 2. Analogy between DS and turbulence

The phenomenon of turbulence appears in many areas of our experience ranging from atmospheric and oceanic rogue events, aero- and hydrodynamics, optics to cardiology and neurophysiology [1, 5, 7–13]. Such a broad class of phenomena cannot be grasped by some single and simple model. However, there are some comparatively simple equations which allow describing an extremely broad class of phenomena. It is possible that the most known one is the famous nonlinear Schrödinger equation (NSE) which describes an evolution of slowly varying wave in a nonlinear medium and can be considered as a "metaphoric" simulation tool for a study of nonlinear phenomena far from equilibrium [14, 15]:

structures including chaos and turbulence is a challenge for modern nonlinear science and one may assume that "the problem of turbulence is one of the central problems in theoretical physics" [1]. The reasonable approach to this issue, which can translate some contra-intuitive and obscure ideas in this area into explicit and verifiable concepts, is a realization of simpler dynamics in quite different material context. Such an approach can be named metaphorical or analog modeling [2], and a rapid progress of modern laser technology provides an ideal playground for such enterprise due to high controllability, relative simplicity, and unique potential of statistic gathering. Such progress was marked by the development of the concept of a dissipative soliton (DS) [3]. The existence of DS under nonequilibrium conditions requires a well-organized energy exchange with an environment so that this energy flow forms a nontrivial internal structure of DS, which provides the energy redistribution inside it and can distort the soliton coherence. Such a DS with nontrivial internal structure can develop in lasers, and the DS dynamics can become chaotic and turbulent [3–5]. For instance, such emergent structures can be considered as a classical analog of Bose-Einstein condensate in low dissipative limit and, contrariwise, as a primitive analog of cell in the case of extensive and wellstructured energy exchange with an environment. Formally, these inherently nonHamiltonian entities mimic some features of Hamiltonian systems that remain an obscure and insufficiently explored topic regarding the fundamental properties of coherent dissipative structures. The range of turbulence, noise, and rogue wave phenomena emulated by the optical DS is so broad that it turns them into a universal testbed for studies in the fields of nonlinear dynamical

98 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

In this work, we conjecture a spectacular analogy between the spectral structures of DS and strong Langmuir turbulence. Such close relation leads to chaotization of DS dynamics with the energy growth. This analogy is deepened by analysis of energy flows inside DS so that a DS can be represented as a "glass of boiling water" or, mathematically, as an ensemble of interacting quasi-particles or "nonlinear modes." The phase decoupling of these "modes" leads to turbulence or DS dissolving. Such a representation open the door for building the kinetic theory of open (dissipative) semi-coherent structures which mimics, in particular, a quantum Bose-Einstein condensate in a dissipative environment. Moreover, our preliminary investigations demonstrated a mechanism of turbulence control provided by noninstantaneous nonlinearity (stimulated Raman scattering in optical case) [6]. This phenomenon is especially interesting because an inherently noisy process (Raman scattering) suppresses a turbulence under some conditions that is the manifestation of stochastic resonance, which can be significant for a dissipation control in coherent quantum systems (particularly, a quantum computer and a

The phenomenon of turbulence appears in many areas of our experience ranging from atmospheric and oceanic rogue events, aero- and hydrodynamics, optics to cardiology and neurophysiology [1, 5, 7–13]. Such a broad class of phenomena cannot be grasped by some single and simple model. However, there are some comparatively simple equations which allow

systems and nonequilibrium thermodynamics.

quantum cryptography device).

2. Analogy between DS and turbulence

$$\frac{\partial \Psi}{\partial T} + \sum\_{j=1}^{d} \frac{\partial \omega}{\partial k\_{j}} \frac{\partial \Psi}{\partial x\_{j}} - \frac{i}{2} \sum\_{j,l=1}^{d} \frac{\partial^{2} \omega}{\partial k\_{j} \partial k\_{l}} \frac{\partial^{2} \Psi}{\partial x\_{j} \partial x\_{l}} + i \left(\frac{\partial \omega}{\partial |\Psi|^{2}}\right) |\Psi|^{2} \Psi = 0. \tag{1}$$

The dimensionality of this equation is relative: the evolutional coordinate can be a time T or a propagation distance z (T\$ z), the transverse coordinate can be transverse multidimensional spatial xj (j = 1…d) one or a local time t (x\$ t, d = 1). The Fourier representations of a "field" slowly varying envelope Ψ are interchangeable between frequency and momentum domains (ω\$ k, d = 1). Eq. (1) may describe the propagation of optical pulses in a nonlinear medium (then Ψ is a complex field amplitude and |Ψ|2 is proportional to a field power), the capillary waves on a fluid surface, the Langmuir waves in plasma, or the weakly nonlinear Bose-gas in classic limit (in the last case Eq. (1) represents the famous time-dependent Gross-Pitaevskii equation [16]).

Here Ψ(x, t) is a slowly varying amplitude of wave propagating in dispersive (P<sup>d</sup> j,l¼1 ∂2 ω ∂kj∂kl ∂2Ψ ∂xj∂xl term; let us <sup>β</sup> � <sup>∂</sup>2<sup>ω</sup> ∂kj∂kl ) and nonlinear ( <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> � �j j <sup>Ψ</sup> <sup>2</sup> <sup>Ψ</sup> - term; let us <sup>γ</sup> � <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> 2) medium. The nonlinear term in Eq. (1) can have the different forms; in particular, a nonlinear response can be noninstantaneous.

The notion of turbulence is fuzzy in some sense. Here, the turbulence will be treated as a phenomenon related to the excitation of a sufficiently large number of degrees of freedom that causes a loss of their mutual phase information [15]. As a consequence, a wave package decouples into a set of individual modes ("particles") which interaction can be described in the framework of kinetic theory as many-particle collisions in Bose-gas. In other words, as some degrees of freedom become very large for sufficiently large energies, phase information becomes irrelevant, and the waves decohere [8, 15]. Thus, a wave can be considered as a set of decoupled "modes" nk in a spectral (or wave-number) space:

$$
\langle \Psi(k)\Psi(k')\rangle = n\_k \delta(k-k').\tag{2}
$$

Thus, we come to a "kinetic" theory of turbulence, for example, to a model of four-boson interaction described by the nonlinear Schrödinger equation:

$$\begin{split} \frac{\partial n\_k}{\partial t} & \propto \left[ (n\_{k\_1} n\_{k\_2} n\_{k\_3} + n\_k n\_{k\_2} n\_{k\_3} - n\_k n\_{k\_1} n\_{k\_2} - n\_k n\_{k\_1} n\_{k\_3}) \right. \\ & \times \delta \left( \overrightarrow{k} + \overrightarrow{k\_1} - \overrightarrow{k\_2} - \overrightarrow{k\_3} \right) \delta (\omega + \omega\_1 - \omega\_2 - \omega\_3) d \overset{\rightarrow}{d} \overrightarrow{k\_1} \; d \; \overrightarrow{k\_2} \; d \; \overrightarrow{k\_3}. \end{split} \tag{3}$$

Such an equation becomes nontrivial in a dissipative environment [17, 18]. A simple generalization of NSE (1) taking into account the dissipative effects includes a saturable gain (energy "source") σ, dissipative nonlinearity (self-amplitude modulation, SAM) Ϝ(|Ψ|2 ), and spectral dissipation (spectral in the sense of dissipation in the Fourier space) P<sup>d</sup> j,l¼<sup>1</sup> <sup>α</sup> xj; xl � � <sup>∂</sup><sup>2</sup> Ψ ∂xj∂xl :

$$\text{RHS of Eq.}(1) = \sigma \Psi + \operatorname{\mathcal{F}}\left(|\Psi|^2\right)\Psi + \sum\_{j\_l=1}^d \alpha(\mathbf{x}\_j, \mathbf{x}\_l) \frac{\partial^2 \Psi}{\partial \mathbf{x}\_j \partial \mathbf{x}\_l}.\tag{4}$$

Eqs. (1) and (4) called the generalized complex nonlinear Ginzburg-Landau equation have the strongly localized (in x-space) steady-state (in T-space) solutions which are named dissipative solitons (DS) [3]. A classical (nondissipative) soliton, which possesses the quite specific mathematical properties [18–20], develops due to mutual compensation of dispersive spreading and self-compression caused by the phase nonlinearity under the condition of <sup>∂</sup>2<sup>ω</sup> ∂kj∂kl � <sup>∂</sup><sup>ω</sup> <sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> <sup>&</sup>gt; 0 and is stable in a (1 + 1)-dimensional (i.e., T plus d = 1 in Eq. (1)) case.<sup>1</sup> The parameters of such soliton are not fixed but only interrelated. One may say that a soliton "lives in solitude" ("pratyekabuddha," Figure 1).

Dissipation adds new bounds on the soliton parameters and fixes them so that one may say that the DS lives in "the heart of nonlinear world" ("bodhisattva," Figure 2).

The mutual balance of dispersion and phase nonlinearity remains a crucial factor for DS formation, but its physical meaning differs substantially from that for nondissipative soliton. The crucial factor here is a resonance between dispersive (linear) waves and DS: equality of their wave-numbers defines the frequency window where DS can exist. Indeed, a wavenumber of DS is <sup>q</sup> <sup>=</sup> <sup>γ</sup>P<sup>0</sup> (P<sup>0</sup> � max(|Ψ|<sup>2</sup> )) [23]. The dispersion relation providing the resonance with linear waves is k(ω) = βω<sup>2</sup> /2. To be stable (i.e., nonradiating), the DS spectrum has to be localized within a frequency window �Δ: <sup>k</sup>(�Δ) = <sup>q</sup>, where <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2γP0=β p (Figure 3).<sup>2</sup> Formation of these "domain walls" [24–26] due to phase effects in a dissipative system results in natural frequency cut-off, which is essential for inherent analogy between DS and a turbulent entity.

However, sole dispersive balance is not sufficient for the DS stability. The spectral dissipation �αΔ<sup>2</sup> plays a crucial role cutting the spectrum and defining the DS width (Figure 4). As

will be seen, this factor is crucial for dissipative soliton turbulence. A spectral dissipation

Figure 4. Wigner representation of DS (a) and cut-off due to spectral dissipation defining the DS width (b).

Figure 3. Resonance conditions for DS and linear waves and DS spectra in dependence on DS energy (a); and the Wigner

the first term is leading) that results in the additional relation between soliton spectral width

condition for the soliton existence which combines the dissipative and nondissipative fac-

The key feature of DS is its nontrivial internal structure revealing itself in the phase inhomo-

∂t <sup>2</sup> .

βκ <sup>≤</sup> <sup>1</sup>=2.<sup>3</sup> One has to note, that both considered mechanisms of DS formation act in the spectral domain and, as was shown in [28], a transition to spectral domain is fruitful for

κP0=α p . In combination with the dispersive relation, it gives the

βκ ≤

1=3 if E ! ∞ 1=2 if E ! 0

) <sup>≈</sup> <sup>κ</sup>|Ψ|<sup>2</sup> � <sup>ζ</sup>|Ψ|<sup>4</sup> <sup>+</sup> …, where

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

101

, where E is a DS energy.

must be balanced by a nonlinear gain �κP<sup>0</sup> (we assume <sup>Ϝ</sup>(|Ψ|<sup>2</sup>

geneity<sup>4</sup> and the internal energy redistribution (E is an energy flow):

More precise analysis [27] gives the conditions of asymptotical stability: αγ

The measure of this inhomogeneity is a so-called chirp <sup>Θ</sup> <sup>∝</sup> <sup>∂</sup><sup>2</sup> argð Þ <sup>Ψ</sup>

and its peak power: <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

(time-spectral) diagram of DS (b) [23].

developing a DS theory.

tors: αγ

3

4

Figure 1. Soliton exists under a balance between phase nonlinearity and dispersion [21, 22].

$$\begin{array}{c} \textbf{phase\ nonlinearity} \longrightarrow \textbf{O} \xleftarrow{\text{losses}} \text{ dispersion} \\ \textbf{!} \text{ gain} \end{array}$$

Figure 2. DS parameters are fixed by both nondissipative and dissipative factors [21, 22].

<sup>1</sup> Further, namely one-dimensional (d = 1) systems will be under consideration that is a quite precise approximation for solid-state and fiber laser dynamics [21].

<sup>2</sup> One has to remind the x \$ t and k\$ ω dualities in Eq. (1).

RHS of Eq:ð Þ¼ <sup>1</sup> <sup>σ</sup><sup>Ψ</sup> <sup>þ</sup> <sup>Ϝ</sup> j j <sup>Ψ</sup> <sup>2</sup> � �

100 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

self-compression caused by the phase nonlinearity under the condition of <sup>∂</sup>2<sup>ω</sup>

that the DS lives in "the heart of nonlinear world" ("bodhisattva," Figure 2).

to be localized within a frequency window �Δ: <sup>k</sup>(�Δ) = <sup>q</sup>, where <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Figure 1. Soliton exists under a balance between phase nonlinearity and dispersion [21, 22].

Figure 2. DS parameters are fixed by both nondissipative and dissipative factors [21, 22].

("pratyekabuddha," Figure 1).

number of DS is <sup>q</sup> <sup>=</sup> <sup>γ</sup>P<sup>0</sup> (P<sup>0</sup> � max(|Ψ|<sup>2</sup>

nance with linear waves is k(ω) = βω<sup>2</sup>

a turbulent entity.

1

2

solid-state and fiber laser dynamics [21].

One has to remind the x \$ t and k\$ ω dualities in Eq. (1).

<sup>Ψ</sup> <sup>þ</sup> <sup>X</sup> d

Eqs. (1) and (4) called the generalized complex nonlinear Ginzburg-Landau equation have the strongly localized (in x-space) steady-state (in T-space) solutions which are named dissipative solitons (DS) [3]. A classical (nondissipative) soliton, which possesses the quite specific mathematical properties [18–20], develops due to mutual compensation of dispersive spreading and

stable in a (1 + 1)-dimensional (i.e., T plus d = 1 in Eq. (1)) case.<sup>1</sup> The parameters of such soliton are not fixed but only interrelated. One may say that a soliton "lives in solitude"

Dissipation adds new bounds on the soliton parameters and fixes them so that one may say

The mutual balance of dispersion and phase nonlinearity remains a crucial factor for DS formation, but its physical meaning differs substantially from that for nondissipative soliton. The crucial factor here is a resonance between dispersive (linear) waves and DS: equality of their wave-numbers defines the frequency window where DS can exist. Indeed, a wave-

Formation of these "domain walls" [24–26] due to phase effects in a dissipative system results in natural frequency cut-off, which is essential for inherent analogy between DS and

However, sole dispersive balance is not sufficient for the DS stability. The spectral dissipation �αΔ<sup>2</sup> plays a crucial role cutting the spectrum and defining the DS width (Figure 4). As

Further, namely one-dimensional (d = 1) systems will be under consideration that is a quite precise approximation for

j, <sup>l</sup>¼<sup>1</sup>

α xj; xl � � <sup>∂</sup><sup>2</sup><sup>Ψ</sup> ∂xj∂xl

: (4)

∂kj∂kl

)) [23]. The dispersion relation providing the reso-

/2. To be stable (i.e., nonradiating), the DS spectrum has

� <sup>∂</sup><sup>ω</sup>

2γP0=β p (Figure 3).<sup>2</sup>

<sup>∂</sup>j j <sup>Ψ</sup> <sup>2</sup> <sup>&</sup>gt; 0 and is

Figure 3. Resonance conditions for DS and linear waves and DS spectra in dependence on DS energy (a); and the Wigner (time-spectral) diagram of DS (b) [23].

Figure 4. Wigner representation of DS (a) and cut-off due to spectral dissipation defining the DS width (b).

will be seen, this factor is crucial for dissipative soliton turbulence. A spectral dissipation must be balanced by a nonlinear gain �κP<sup>0</sup> (we assume <sup>Ϝ</sup>(|Ψ|<sup>2</sup> ) <sup>≈</sup> <sup>κ</sup>|Ψ|<sup>2</sup> � <sup>ζ</sup>|Ψ|<sup>4</sup> <sup>+</sup> …, where the first term is leading) that results in the additional relation between soliton spectral width and its peak power: <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi κP0=α p . In combination with the dispersive relation, it gives the condition for the soliton existence which combines the dissipative and nondissipative factors: αγ βκ <sup>≤</sup> <sup>1</sup>=2.<sup>3</sup> One has to note, that both considered mechanisms of DS formation act in the spectral domain and, as was shown in [28], a transition to spectral domain is fruitful for developing a DS theory.

The key feature of DS is its nontrivial internal structure revealing itself in the phase inhomogeneity<sup>4</sup> and the internal energy redistribution (E is an energy flow):

<sup>3</sup> More precise analysis [27] gives the conditions of asymptotical stability: αγ βκ ≤ 1=3 if E ! ∞ 1=2 if E ! 0 , where E is a DS energy.

<sup>4</sup> The measure of this inhomogeneity is a so-called chirp <sup>Θ</sup> <sup>∝</sup> <sup>∂</sup><sup>2</sup> argð Þ <sup>Ψ</sup> ∂t <sup>2</sup> .

$$E \equiv \frac{i}{2} \frac{\partial}{\partial t} \left( \Psi \frac{\partial \Psi^\*}{\partial t} - \Psi^\* \frac{\partial \Psi}{\partial t} \right) = 2\sigma |\Psi|^2 + 2\kappa |\Psi|^4 - 2\alpha \left| \frac{\partial \Psi}{\partial t} \right| + \alpha \frac{\partial^2 |\Psi|^2}{\partial t^2} . \tag{5}$$

The third term in RHS of Eq. (5) is phase-sensitive and, thus, there is an energy flow from DS center, where spectral components with minimal relative frequencies are located, to the DS wings, where frequency components with maximum relative frequencies are located (Figure 5). Here, energy dissipates. Such nontrivial internal "life" of DS intensifies with the growth of phase inhomogeneity Θ. Simultaneously, DS becomes an energy scalable coherent concentrate with the energy ("concentrate mass") ∝Θ [5].

As a result of phase inhomogeneity and intensive internal energy flows, the internal coherence of DS can become partially broken. Then, DS splits into partially coherent "internal modes" which interact with each other as the independent "sub-solitons." [29–31] Thus, DS becomes a strongly localized "cloud" of interacting "quasi-particles" or "glass of boiling water" (Figure 6).

These figures demonstrate an affinity between the structures of DS and turbulence [8, 23]. Both spectral structures are defined by dispersion relations: between soliton and dispersive waves for the former and Langmuir dispersion relation for the latter (Figure 7). Secondly, both highenergy DS and turbulence are characterized by spectral condensation at zero frequency (wavenumber) with subsequent scattering to higher frequencies confined by cut-off at �Δ.

Such an analogy between DS and turbulence opens a door for building the kinetic and quantum [5, 32–34] theory of open (dissipative) semi-coherent structures which mimics, in particular, a quantum Bose-Einstein condensate in a dissipative environment.

3. Transition to a DS turbulence

The value ∝1/Ξ can be treated as a measure of "long-range" correlation scale.

The value ∝1/Δ can be treated as a measure of "short-range" correlation scale.

dissipation (adapted from [8]).

<sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

5

6

The mechanism of transition to turbulence for DS can be associated with the time/spectral duality (Figure 8). When the energy increases (i.e., E ! ∞ that corresponds to a system with "infinite capacity" [36]), the spectrum condensates around ω = 0 within a diapason of Ξ ! 0 (Figure 7). Simultaneously, DS broadens in time domain ∝1/Ξ by analogy with a growth of Bose-Einstein condensate "mass."<sup>5</sup> The DS peak power tends to some constant value P0∝1/ζ defined by a saturation of dissipative nonlinearity (see above), and, thereby, the cut-off frequency

Figure 7. DS spectrum (central red curve in the left picture) where the cut-off frequency Δ is defined by the resonance condition between linear waves with a wave-number k and DS with a wave-number q (parabolic black curve in the left picture; P<sup>0</sup> is a DS peak power) [23]. The turbulence in the wave-number space is defined by the Langmuir dispersion relation (parabolic black curve in the right picture). Spectral condensation at k = 0 is illustrated by shading and forms a characteristic turbulence spectrum (central red curve; right picture) with the cut-off wave-number �kdiss defined by a

Figure 6. DS spectrum as a "glass of boiling water" (left) [30] demonstrating the dynamics of internal perturbation modes (right) [31]. The last picture is obtained by a perturbation analysis in spectral domain with the Neumann series expansion.

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

103

κP0=α p tends to be constant.  Energy flows (left column) and corresponding spectral profiles (right column, ℱ is a Fourier image of Ψ) in dependence on dispersion β and chirp Θ for a DS with the profile Ψ ∝sech(t) 1 + <sup>i</sup><sup>Θ</sup> [4].

Figure 6. DS spectrum as a "glass of boiling water" (left) [30] demonstrating the dynamics of internal perturbation modes (right) [31]. The last picture is obtained by a perturbation analysis in spectral domain with the Neumann series expansion.

Figure 7. DS spectrum (central red curve in the left picture) where the cut-off frequency Δ is defined by the resonance condition between linear waves with a wave-number k and DS with a wave-number q (parabolic black curve in the left picture; P<sup>0</sup> is a DS peak power) [23]. The turbulence in the wave-number space is defined by the Langmuir dispersion relation (parabolic black curve in the right picture). Spectral condensation at k = 0 is illustrated by shading and forms a characteristic turbulence spectrum (central red curve; right picture) with the cut-off wave-number �kdiss defined by a dissipation (adapted from [8]).

#### 3. Transition to a DS turbulence

<sup>E</sup> � <sup>i</sup> 2 ∂ ∂t <sup>Ψ</sup> <sup>∂</sup>Ψ<sup>∗</sup>

with the energy ("concentrate mass") ∝Θ [5].

<sup>∂</sup><sup>t</sup> � <sup>Ψ</sup><sup>∗</sup> <sup>∂</sup><sup>Ψ</sup> ∂t

<sup>¼</sup> <sup>2</sup>σj j <sup>Ψ</sup> <sup>2</sup> <sup>þ</sup> <sup>2</sup>κj j <sup>Ψ</sup> <sup>4</sup> � <sup>2</sup><sup>α</sup>

The third term in RHS of Eq. (5) is phase-sensitive and, thus, there is an energy flow from DS center, where spectral components with minimal relative frequencies are located, to the DS wings, where frequency components with maximum relative frequencies are located (Figure 5). Here, energy dissipates. Such nontrivial internal "life" of DS intensifies with the growth of phase inhomogeneity Θ. Simultaneously, DS becomes an energy scalable coherent concentrate

As a result of phase inhomogeneity and intensive internal energy flows, the internal coherence of DS can become partially broken. Then, DS splits into partially coherent "internal modes" which interact with each other as the independent "sub-solitons." [29–31] Thus, DS becomes a strongly

These figures demonstrate an affinity between the structures of DS and turbulence [8, 23]. Both spectral structures are defined by dispersion relations: between soliton and dispersive waves for the former and Langmuir dispersion relation for the latter (Figure 7). Secondly, both highenergy DS and turbulence are characterized by spectral condensation at zero frequency (wavenumber) with subsequent scattering to higher frequencies confined by cut-off at �Δ. Such an analogy between DS and turbulence opens a door for building the kinetic and quantum [5, 32–34] theory of open (dissipative) semi-coherent structures which mimics, in

Figure 5. Energy flows (left column) and corresponding spectral profiles (right column, ℱ is a Fourier image of Ψ) in

1 + <sup>i</sup><sup>Θ</sup> [4].

dependence on dispersion β and chirp Θ for a DS with the profile Ψ ∝sech(t)

localized "cloud" of interacting "quasi-particles" or "glass of boiling water" (Figure 6).

particular, a quantum Bose-Einstein condensate in a dissipative environment.

∂Ψ ∂t

 þ α <sup>∂</sup><sup>2</sup>j j <sup>Ψ</sup> <sup>2</sup> ∂t

<sup>2</sup> : (5)

102 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

The mechanism of transition to turbulence for DS can be associated with the time/spectral duality (Figure 8). When the energy increases (i.e., E ! ∞ that corresponds to a system with "infinite capacity" [36]), the spectrum condensates around ω = 0 within a diapason of Ξ ! 0 (Figure 7). Simultaneously, DS broadens in time domain ∝1/Ξ by analogy with a growth of Bose-Einstein condensate "mass."<sup>5</sup> The DS peak power tends to some constant value P0∝1/ζ defined by a saturation of dissipative nonlinearity (see above), and, thereby, the cut-off frequency <sup>Δ</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi κP0=α p tends to be constant. The last value defines the coherence scale ∝1/Δ (few picoseconds for a typical DS).6 As a result, DS becomes "decoupled," and even small perturbations can

<sup>5</sup> The value ∝1/Ξ can be treated as a measure of "long-range" correlation scale.

<sup>6</sup> The value ∝1/Δ can be treated as a measure of "short-range" correlation scale.

Figure 8. DS spectrum (red curves) which cut-off frequency Δ defines the correlation scale ∝1/Δ. It tends to some constant value with the DS energy growth. Simultaneously, DS broadens in time domain ∝1/Ξ (blue curves) in parallel with the concentration of energy around ω = 0 in spectral domain [35].

destroy its internal coherence and split ("nucleate") it into a set of "internal modes" shown in Figure 6.

More close insight into this mechanism can be provided by the adiabatic theory of DS in spectral domain presented in [28]. As was shown, the DS spectrum can be expressed as follows:

$$p(\omega) = \frac{\Upsilon}{\omega^2 + \Xi^2} H(\Delta^2 - \omega^2),\tag{6}$$

characterized by the short-range correlation time about of 1 ps. In the process of evolution, an incoherent DS splits into three almost identical coherent solitons, which widths are lower substantially and the corresponding long-range/short-range correlation times become smaller/larger, respectively. Small long-range correlation time prevents the DSs merging and

Figure 9. Contour plot of the DS power evolution and the corresponding autocorrelation function on the turbulent stage. The evolution coordinate is measured in the laser cavity round-trips, and the transverse coordinate corresponds to the

fused silica nonlinear coefficient and Δt=1 femtosecond is a time discretization step. Other parameters are: κ = 0.1γ,

, where γ corresponds to a

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

105

local time measured in picoseconds. The energy parameter Es normalized on <sup>γ</sup>/Δ<sup>t</sup> is of 3 � 105

An analysis of turbulent DS demonstrates its complicate internal structure which can be interpreted as the complex of strongly interacting bright, dark, and gray DSs on a finite but strongly self-localized background concentrating almost the entire part of the energy.<sup>8</sup> In some sense, an appearance of DS turbulence resembles the laminar-turbulent transition in a fiber laser when a macroscopically coherent field (Ξ!0) becomes chaotically self-modulated [45]. Nevertheless, such a scenario is not unique. The turbulent dynamics can result from the strong interaction between "individual" DSs forming a "soliton gas" or turbulent "soliton cluster" (Figure 10) [1, 5, 47]. Interaction of such cluster with a low-intensity background field can result in permanent radiation or absorption of DSs in the form of so-called "soliton rains" [48, 49].

Separately, one may note the chaotization of DS dynamics caused by resonant interaction with the dispersive waves in the presence of higher-order corrections to the dispersion term in Eq. (1). In this case, the collisions between DS and dispersive wave, which radiates by it, results

One has to distinguish such a structure from the breather-like structures on a continuous-wave background. Such

in a chaotic dynamic preserving, nevertheless the DS integrity (Figure 11) [23].

structures can demonstrate chaotic and rogue waves dynamics, as well (e.g., see [43, 44]).

larger short-range correlation time provides DS coherence.

ζ = 0.05γ, δ = 0.05, and α = 40 nm (Yb-fiber laser) [6].

8

where p(ω) is a DS spectral power, and H is the Heaviside function. Eq. (6) represents the spectra shown in Figures 7 and 8, and can be interpreted by analogy with the Rayleigh-Jeans distribution, so that Ξ<sup>2</sup> plays a role of negative "chemical potential" [8, 36, 37]. The parameter Υ = 6πγ/κζ is an analog of "temperature" and is defined by both dissipative and nondissipative nonlinear parameters.

Since the "chemical potential" Ξ<sup>2</sup> decays with the energy growth (Figure 8), a system tends to the state of "soliton gas" [38] with the characteristic "soliton size" ∝1/Δ. Thereby, a coherent "condensate" with minimum entropy becomes a state of the decomposed "quasi-particles" with the chaotically modulated powers because the required entropy growth is provided by such modulation<sup>7</sup> [40]. Thus, the energy growth (i.e., the growth of "condensate mass" ∝1/Ξ) leads to extra-sensitivity to quantum-level noises [40, 41] that urges the quantum theory of coherent and semi-coherent dissipative structures, which would weave largest and smallest scales in the DS dynamics.

The example of such "DS decomposition" through a turbulence is shown in Figure 9. This figure is obtained by numerical simulation of Eqs. (1) and (4) with taking into account of the gain saturation in the form of σ = δ(1 � E/Es) [27]. Figure 9 demonstrates clearly two stages of DS evolution. The first stage corresponds to an incoherent and strongly turbulent DS, which is

<sup>7</sup> Here, one may draw an analogy with Hamiltonian systems, where the gradient of field is a measure of the amount of fluctuations [39].

Self-Organization, Coherence and Turbulence in Laser Optics http://dx.doi.org/10.5772/intechopen.71478 105

Figure 9. Contour plot of the DS power evolution and the corresponding autocorrelation function on the turbulent stage. The evolution coordinate is measured in the laser cavity round-trips, and the transverse coordinate corresponds to the local time measured in picoseconds. The energy parameter Es normalized on <sup>γ</sup>/Δ<sup>t</sup> is of 3 � 105 , where γ corresponds to a fused silica nonlinear coefficient and Δt=1 femtosecond is a time discretization step. Other parameters are: κ = 0.1γ, ζ = 0.05γ, δ = 0.05, and α = 40 nm (Yb-fiber laser) [6].

destroy its internal coherence and split ("nucleate") it into a set of "internal modes" shown in

Figure 8. DS spectrum (red curves) which cut-off frequency Δ defines the correlation scale ∝1/Δ. It tends to some constant value with the DS energy growth. Simultaneously, DS broadens in time domain ∝1/Ξ (blue curves) in parallel with the

More close insight into this mechanism can be provided by the adiabatic theory of DS in spectral

where p(ω) is a DS spectral power, and H is the Heaviside function. Eq. (6) represents the spectra shown in Figures 7 and 8, and can be interpreted by analogy with the Rayleigh-Jeans distribution, so that Ξ<sup>2</sup> plays a role of negative "chemical potential" [8, 36, 37]. The parameter Υ = 6πγ/κζ is an analog of "temperature" and is defined by both dissipative and nondissipative

Since the "chemical potential" Ξ<sup>2</sup> decays with the energy growth (Figure 8), a system tends to the state of "soliton gas" [38] with the characteristic "soliton size" ∝1/Δ. Thereby, a coherent "condensate" with minimum entropy becomes a state of the decomposed "quasi-particles" with the chaotically modulated powers because the required entropy growth is provided by such modulation<sup>7</sup> [40]. Thus, the energy growth (i.e., the growth of "condensate mass" ∝1/Ξ) leads to extra-sensitivity to quantum-level noises [40, 41] that urges the quantum theory of coherent and semi-coherent dissipative structures, which would weave largest and smallest

The example of such "DS decomposition" through a turbulence is shown in Figure 9. This figure is obtained by numerical simulation of Eqs. (1) and (4) with taking into account of the gain saturation in the form of σ = δ(1 � E/Es) [27]. Figure 9 demonstrates clearly two stages of DS evolution. The first stage corresponds to an incoherent and strongly turbulent DS, which is

Here, one may draw an analogy with Hamiltonian systems, where the gradient of field is a measure of the amount of

<sup>ω</sup><sup>2</sup> <sup>þ</sup> <sup>Ξ</sup><sup>2</sup> <sup>H</sup> <sup>Δ</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup> , (6)

domain presented in [28]. As was shown, the DS spectrum can be expressed as follows:

Υ

pð Þ¼ ω

concentration of energy around ω = 0 in spectral domain [35].

104 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

Figure 6.

nonlinear parameters.

scales in the DS dynamics.

7

fluctuations [39].

characterized by the short-range correlation time about of 1 ps. In the process of evolution, an incoherent DS splits into three almost identical coherent solitons, which widths are lower substantially and the corresponding long-range/short-range correlation times become smaller/larger, respectively. Small long-range correlation time prevents the DSs merging and larger short-range correlation time provides DS coherence.

An analysis of turbulent DS demonstrates its complicate internal structure which can be interpreted as the complex of strongly interacting bright, dark, and gray DSs on a finite but strongly self-localized background concentrating almost the entire part of the energy.<sup>8</sup> In some sense, an appearance of DS turbulence resembles the laminar-turbulent transition in a fiber laser when a macroscopically coherent field (Ξ!0) becomes chaotically self-modulated [45].

Nevertheless, such a scenario is not unique. The turbulent dynamics can result from the strong interaction between "individual" DSs forming a "soliton gas" or turbulent "soliton cluster" (Figure 10) [1, 5, 47]. Interaction of such cluster with a low-intensity background field can result in permanent radiation or absorption of DSs in the form of so-called "soliton rains" [48, 49].

Separately, one may note the chaotization of DS dynamics caused by resonant interaction with the dispersive waves in the presence of higher-order corrections to the dispersion term in Eq. (1). In this case, the collisions between DS and dispersive wave, which radiates by it, results in a chaotic dynamic preserving, nevertheless the DS integrity (Figure 11) [23].

<sup>8</sup> One has to distinguish such a structure from the breather-like structures on a continuous-wave background. Such structures can demonstrate chaotic and rogue waves dynamics, as well (e.g., see [43, 44]).

U tðÞ¼ <sup>T</sup><sup>2</sup>

discharging" is like the turbulence decay shown in Figure 9.

Figure 13. Wigner (time-spectral) diagram of a sole DRS developing for large β [6].

nonlinear medium.

power; right) [35].

<sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> 2 T1T<sup>2</sup> 2

exp � <sup>t</sup> T2 sin <sup>t</sup>

Where T<sup>2</sup> and T<sup>1</sup> define the effective relaxation time and resonant frequency of phonons in a

The simulations demonstrate [6, 35] that SRS suppresses the DS turbulence for the sufficiently large dispersions β. The first scenario is formation of uncoupled complex of DS and the dissipative Raman soliton (DRS) [6, 35, 42, 52] (Figure 12). One may assume, that such "energy

DRS is characterized by large chirp Θ and frequency down-shift. The last results from intrapulse SRS which is possible due to a broad spectrum, which is a common characteristic of DS and results from its large Θ. A sole DRS develops with growing β (Figure 13) [6, 35]. It is

Figure 12. Wigner (time-spectral) diagram of the DS + DRS complex (left) and its evolution (contour plot of the field

T1

, (8)

http://dx.doi.org/10.5772/intechopen.71478

107

Self-Organization, Coherence and Turbulence in Laser Optics

Figure 10. DS clusters in the form of a "persistent and coherent quasi-soliton" (left) [46] and a "sporadic rogue waves events that emerge from turbulent fluctuations" (right) [41].

Figure 11. Wigner (time-spectral) diagram of the chaotic DS in the presence of third-order dispersion [23]. DS (dark-red region around 2.3 µm) radiates a dispersive wave (blue tail around 2.4 µm). As a result of the difference between their group velocities, DS collides permanently with a dispersive wave that causes chaotization of dynamics and modulation of both DS spectrum and time-profile.

#### 4. Coherence of DS in the presence of nonlinearity with nonlocal/noninstantaneous response

A nonlinearity with the nonlocal/noninstanteneous response, which is of interest in optical context, can be taken into account by inclusion in Eq. (1) of the following term [50]

$$-i\gamma^{\mu}\Psi\int \mathcal{U}(\mathbf{x}-\mathbf{x}')|\Psi|^2(T,\mathbf{x}')d\mathbf{x}'.\tag{7}$$

In the case of nonstationarity (i.e., x !t replacement), this equation describes the stimulated Raman scattering (SRS), for instance. Then, the response function is [6, 51]:

$$\Delta U(t) = \frac{T\_1^2 + T\_2^2}{T\_1 T\_2^2} \exp\left(-\frac{t}{T\_2}\right) \sin\left(\frac{t}{T\_1}\right),\tag{8}$$

Where T<sup>2</sup> and T<sup>1</sup> define the effective relaxation time and resonant frequency of phonons in a nonlinear medium.

The simulations demonstrate [6, 35] that SRS suppresses the DS turbulence for the sufficiently large dispersions β. The first scenario is formation of uncoupled complex of DS and the dissipative Raman soliton (DRS) [6, 35, 42, 52] (Figure 12). One may assume, that such "energy discharging" is like the turbulence decay shown in Figure 9.

DRS is characterized by large chirp Θ and frequency down-shift. The last results from intrapulse SRS which is possible due to a broad spectrum, which is a common characteristic of DS and results from its large Θ. A sole DRS develops with growing β (Figure 13) [6, 35]. It is

Figure 12. Wigner (time-spectral) diagram of the DS + DRS complex (left) and its evolution (contour plot of the field power; right) [35].

Figure 13. Wigner (time-spectral) diagram of a sole DRS developing for large β [6].

4. Coherence of DS in the presence of nonlinearity with

�iγΨ ð

Raman scattering (SRS), for instance. Then, the response function is [6, 51]:

A nonlinearity with the nonlocal/noninstanteneous response, which is of interest in optical

Figure 11. Wigner (time-spectral) diagram of the chaotic DS in the presence of third-order dispersion [23]. DS (dark-red region around 2.3 µm) radiates a dispersive wave (blue tail around 2.4 µm). As a result of the difference between their group velocities, DS collides permanently with a dispersive wave that causes chaotization of dynamics and modulation of

Figure 10. DS clusters in the form of a "persistent and coherent quasi-soliton" (left) [46] and a "sporadic rogue waves

In the case of nonstationarity (i.e., x !t replacement), this equation describes the stimulated

U x � <sup>x</sup><sup>0</sup> ð Þj j <sup>Ψ</sup> <sup>2</sup> <sup>T</sup>; <sup>x</sup><sup>0</sup> ð Þdx<sup>0</sup>

: (7)

context, can be taken into account by inclusion in Eq. (1) of the following term [50]

nonlocal/noninstantaneous response

both DS spectrum and time-profile.

events that emerge from turbulent fluctuations" (right) [41].

106 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

turbulence-free and characterized by perturbed anti-Stokes component, which is clearly visible on the Wigner diagram. Such perturbation induces a chaotic vibration of the DRS power [35]. Nevertheless, DRS exists within the parametric range, where an ordinary DS cannot develop in the turbulence-free regime. One may assume that the DRS stability results from passive negative feed-back based on interplay between nonlinear down-frequency shift due to SRS and spectral dissipation.

Acknowledgements

Author details

References

2013.359

10.1007/b11728

2012.01.004

tific; 2013

Vladimir L. Kalashnikov\* and Evgeni Sorokin

\*Address all correspondence to<EMAIL_ADDRESS>
2004;398:1-65. DOI: 10.1016/j.physrep.2004.04.002

Fiber Laser. Rijeka: InTechOpen; 2016. p. 165-210

and Nonlinear Media. New Jersey: World Scientific; 2006

ics. 1997;69:507-574. DOI: 10.1103/RevModPhys.69.507

Chapman & Hall; 2016. p. 587-626

30126. DOI: 10.1364/OE.22.030118

Institute of Photonics, Vienna University of Technology, Vienna, Austria

The author acknowledges the support from Austrian Science Fund (FWF Project No. P24916- N27). Computational results have been achieved using the Vienna Scientific Cluster (VSC).

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

109

[1] Zakharov V, Diasc F, Pushkarev A. One-dimensional wave turbulence. Physics Reports.

[2] Editorial. The power of analogies. Nature Photonics. 2014;8:1. DOI: 10.1038/nphoton.

[3] Akhmediev N, Ankiewicz A, editors. Dissipative Solitons. Berlin: Springer; 2005. DOI:

[4] Kalashnikov VL, Sergeyev SV. Dissipative solitons in fibre lasers. In: Paul MC, editor.

[5] Kalashnikov VL. Optics and chaos: Chaotic, rogue and noisy optical dissipative Solitons. In: Skiadas CH, Skiadas C, editors. Handbook of Applications of Chaos Theory. London:

[6] Kalashnikov VL, Sorokin E. Dissipative Raman solitons. Optics Express. 2014;22:30118-

[7] Shats M, Punzmann H, editors. Turbulence and Coherent Structures in Fluids, Plasmas

[8] Robinson PA. Nonlinear wave collapse and strong turbulence. Reviews of Modern Phys-

[9] Laurie J, Bortolozzo U, Nazarenko S, Residori S. One-dimensional optical wave turbulence: Experiment and theory. Physics Reports. 2012;514:121-175. DOI: 10.1016/j.physrep.

[10] Shrira V, Nazarenko S, editors. Advances in Wave Turbulence. New Jersey: World Scien-

Another spectacular manifestation of the effect of a noninstantaneous nonlinearity on an incoherent field is an appearance of the spectral incoherent solitons (SIS) [53]. The spectrally localized soliton-like structures appear without any time-localization due to the causality property inherent to SRS so that a field cannot reach thermal equilibrium [54]. Formally, the corresponding evolution equation in the Langmuir turbulence limit has soliton-like solutions in the spectral domain [55, 56]. As a result, such structure localized in spectral domain possesses main properties of solitons including the property of elastic scattering.

#### 5. Conclusions

The problem of DS coherence, chaotic, and turbulent dynamics has been outlined. A nontrivial internal structure of DS caused by it intensive energy exchange with dissipative environment allows conjecturing a close analogy with turbulent structure forming far from equilibrium. The existence of long-range correlation scale provides the DS energy scaling (or mass scaling for Bose-Einstein condensate). However, such "macroscopic" scaling is provided by strong phase inhomogeneity (chirp) so that internal coherence of DS defined by short-range correlation scale breaks and DS becomes a "cloud" of interacting "quasi-particles" or "glass of boiling water." Such structure is very sensitive to perturbation of even quantum level. Such extra-sensitivity combines macro- and micro-scales that raises an issue of the quantum theory of the macroscopic coherent, partially, and incoherent dissipative structures.

In the context of this work, such DS "decomposition" leads to turbulent dynamics and DS fragmentation. In particular, interactions inside such "soliton cluster" can result in the rogue waves' formation. An additional source of soliton destabilization is resonant interaction with a dispersive wave that results in chaotization of dynamics and formation of "soliton rains."

Nonlinearity with noninstanteneous response (e.g., SRS) leads to new interesting effect. In particular, SRS suppresses the DS turbulence due to the formation of DS + DRS pairs or sole DRS. Although DRS is turbulence-free within a broad parametric range, it has a perturbed anti-Stokes component, which causes chaotic vibrations of DRS parameters.

Another and spectacular manifestation of the noninstantaneous response of nonlinearity is the formation of SIS. This structure is a soliton in the spectral domain but incoherent and delocalized in the time domain.

The above-considered phenomena and conjectures are of interest in the context of the development of approaches to the self-consistent theory of nonequilibrium dissipative structures in classical and quantum aspects, which would use the optical DSs as a testbed.

#### Acknowledgements

turbulence-free and characterized by perturbed anti-Stokes component, which is clearly visible on the Wigner diagram. Such perturbation induces a chaotic vibration of the DRS power [35]. Nevertheless, DRS exists within the parametric range, where an ordinary DS cannot develop in the turbulence-free regime. One may assume that the DRS stability results from passive negative feed-back based on interplay between nonlinear down-frequency shift due to SRS

Another spectacular manifestation of the effect of a noninstantaneous nonlinearity on an incoherent field is an appearance of the spectral incoherent solitons (SIS) [53]. The spectrally localized soliton-like structures appear without any time-localization due to the causality property inherent to SRS so that a field cannot reach thermal equilibrium [54]. Formally, the corresponding evolution equation in the Langmuir turbulence limit has soliton-like solutions in the spectral domain [55, 56]. As a result, such structure localized in spectral domain pos-

The problem of DS coherence, chaotic, and turbulent dynamics has been outlined. A nontrivial internal structure of DS caused by it intensive energy exchange with dissipative environment allows conjecturing a close analogy with turbulent structure forming far from equilibrium. The existence of long-range correlation scale provides the DS energy scaling (or mass scaling for Bose-Einstein condensate). However, such "macroscopic" scaling is provided by strong phase inhomogeneity (chirp) so that internal coherence of DS defined by short-range correlation scale breaks and DS becomes a "cloud" of interacting "quasi-particles" or "glass of boiling water." Such structure is very sensitive to perturbation of even quantum level. Such extra-sensitivity combines macro- and micro-scales that raises an issue of the quantum theory of the macro-

In the context of this work, such DS "decomposition" leads to turbulent dynamics and DS fragmentation. In particular, interactions inside such "soliton cluster" can result in the rogue waves' formation. An additional source of soliton destabilization is resonant interaction with a dispersive wave that results in chaotization of dynamics and formation of "soliton rains."

Nonlinearity with noninstanteneous response (e.g., SRS) leads to new interesting effect. In particular, SRS suppresses the DS turbulence due to the formation of DS + DRS pairs or sole DRS. Although DRS is turbulence-free within a broad parametric range, it has a perturbed

Another and spectacular manifestation of the noninstantaneous response of nonlinearity is the formation of SIS. This structure is a soliton in the spectral domain but incoherent and

The above-considered phenomena and conjectures are of interest in the context of the development of approaches to the self-consistent theory of nonequilibrium dissipative structures in

anti-Stokes component, which causes chaotic vibrations of DRS parameters.

classical and quantum aspects, which would use the optical DSs as a testbed.

sesses main properties of solitons including the property of elastic scattering.

108 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

scopic coherent, partially, and incoherent dissipative structures.

and spectral dissipation.

5. Conclusions

delocalized in the time domain.

The author acknowledges the support from Austrian Science Fund (FWF Project No. P24916- N27). Computational results have been achieved using the Vienna Scientific Cluster (VSC).

#### Author details

Vladimir L. Kalashnikov\* and Evgeni Sorokin

\*Address all correspondence to<EMAIL_ADDRESS>
Institute of Photonics, Vienna University of Technology, Vienna, Austria

#### References

[28] Podivilov E, Kalashnikov VL. Heavily-chirped solitary pulses in the normal dispersion region: New solutions of the cubic-quintic Ginzburg-Landau equation. JETP Letters.

Self-Organization, Coherence and Turbulence in Laser Optics

http://dx.doi.org/10.5772/intechopen.71478

111

[29] Akhmediev N, Królikowski W, Snyder AW. Partially coherent solitons of variable shape. Physical Review Letters. 1998;81:4632-4635. DOI: 10.1103/PhysRevLett.81.4632

[30] Kalashnikov VL, Chernykh A. Spectral anomalies and stability of chirped-pulse oscilla-

[31] Kalashnikov VL. Dissipative solitons: Perturbations and chaos formation. In: III Chaotic Modelling and Simulation International Conference; June 1–4, 2010; Chania, Greece;

[32] Yoon B, Negele JW. Time-dependent approximation for a one-dimensional system of bosons with attractive δ-function interactions. Physical Review A. 1977;16:1451-1457.

[33] Lai Y, Haus HA. Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation. Physical Review A. 1989;40:844-853. DOI: 10.1103/PhysRevA.40.844 [34] Lai Y, Haus HA. Quantum theory of solitons in optical fibers. II. Exact solution. Physical

[35] Kalashnikov VL, Sorokin E. Turbulence of optical dissipative solitons. In: X Chaotic Modelling and Simulation International Conference; May 30–June 2, 2017; Barcelona,

[36] Newell AC, Zakharov VE. Optical turbulence. In: Tabeling P, Cardoso O, editors. Turbu-

[37] Picozzi A. Towards a nonequilibrium thermodynamic description of incoherent nonlinear

[38] D'yachenko AI, Zakharov VE, Pushkarev AN, Shvets VF, Yan'kov VV. Soliton turbulence

[39] Düring G, Picozzi A, Rica S. Breakdown of weak-turbulence and nonlinear wave condensation. Physica D: Nonlinear Phenomena. 2009;238:1524-1549. DOI: 10.106/j.physd.2009.

[40] Pomeau Y. Asymptotic time behavior of nonlinear classical field equations. Nonlinearity.

[41] Kalashnikov VL. Dissipative solitons in presence of quantum noise. Chaotic Modeling

[42] Babin SA, Podivilov EV, Kharenko DS, Bednyakova AE, Fedoruk MP, Shtyrina OV, Kalashnikov VL, Apolonski AA. SRS-driven evolution of dissipative solitons in fiber lasers. In: Grelu P, editor. Nonlinear Optical Cavity Dynamics: From Microresonators to

lence: A Tentative Dictionary. New York: Plenum Press; 1993. p. 59-66

in nonintegrable wave systems. Soviet Physics—JETP. 1990;69:1144-1147

Review A. 1989;40:854-866. DOI: 10.1103/PhysRevA.40.854

tors. Physical Review A. 2007;75:033820. DOI: 10.1103/PhysRevA.75.033820

2005;82:467-471

2010.

Spain; 2017.

04.014

1992;5:707-720

DOI: 10.1103/PhysRevA.16.1451

optics. Optics Express. 2007;15:9063-2083

and Simulation (CMSIM). 2014:29-37

Fiber Optics. Weinheim: Wiley-VCH; 2016. p. 277-346

[28] Podivilov E, Kalashnikov VL. Heavily-chirped solitary pulses in the normal dispersion region: New solutions of the cubic-quintic Ginzburg-Landau equation. JETP Letters. 2005;82:467-471

[11] Nazarenko S. Wave Turbulence. Berlin: Springer; 2011

110 Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

[13] Ilday FÖ. Turbulent times. Nature Photonics. 2013;7:767-769

Mathematics. 1967;46:133-139. DOI: 10.1002/sapm1967461133

Phenomena in Bose-Einstein Condensates. Berlin: Springer; 2008

Nature Communications. 2015;6:6214. DOI: 10.1038/ncomms7214

Medicine. Berlin: Springer; 2008. DOI: 10.1007/978-3-540-78217-9

Optics Express. 2013;21:29567-29577. DOI: 10.1364/OE.21.029567

[18] Zakharov VE, editor. What is Integrability? Berlin: Springer; 1991

2012;6:84-92. DOI: 10.1038/NPHOTON.2011.345

1990

62-69

143906

PhysRevA.83.033838

& Hall; 1997

[12] Manneville P. Dissipative Structures and Weak Turbulence. San Diego: Academic Press;

[14] Benney DJ, Newell AC. The propagation of nonlinear wave envelopes. Studies in Applied

[15] Dyachenko S, Newell AC, Pushkarev A, Zakharov VE. Optical turbulence: Weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D: Nonlinear Phenomena. 1992;57:96-160. DOI: 10.1016/0167-2789(92)90090-A

[16] Kevrekidis PG, Frantzeskakis DJ, Carretero-González R, editors. Emergent Nonlinear

[17] Churkin DV, Kolokolov IV, Podivilov EV, Vatnik ID, Nikulin MA, Vergeles SS, Terekhov S, Lebedev VV, Falkovich G, Babin SA, Turitsyn SK. Wave kinetics of random fibre lasers.

[19] Zakharov VE, Shabat AB. Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media. Soviet Physics—JETP. 1972;34:

[20] Akhmediev NN, Ankiewicz A. Solitons: Nonlinear Pulses and Beams. London: Chapman

[21] Grelu P, Akhmediev N. Dissipative solitons for mode-locked lasers. Nature Photonics.

[22] Akhmediev N, Ankiewicz A, editors. Dissipative Solitons: From Optics to Biology and

[23] Sorokin E, Tolstik N, Kalashnikov VL, Sorokina IT. Chaotic chirped-pulse oscillators.

[24] Rumpf B, Newell AC. Localization and coherence in nonintegrable systems. Physica D: Nonlinear Phenomena. 2003;184:162-191. DOI: 10.1016/S0167-2789(03)00220-3

[25] Picozzi A, Haeltermann M, Pitois S, Millot G. Incoherent solitons in instantaneous response nonlinear media. Physical Review Letters. 2004;92:143906. DOI: 10.1103/PhysRevLett.92.

[26] Aschieri P, Garnier J, Michel C, Doya V, Picozzi A. Condensation and thermalization of classical optical waves in a waveguide. Physical Review A. 2011;83:033838. DOI: 10.1103/

[27] Kalashnikov VL. Chirped-pulse oscillators: Route to the energy scalable femtosecond pulses. In: Al-Khursan AH, editor. Solid-State Laser. Rijeka: InTechOpen; 2012. p. 145-184

[43] Onorato M, Residori S, Baronio F, editors. Rogue and Shock Waves in Nonlinear Dispersive Media. Switzerland: Springer; 2016. DOI: 10.1007/978-3-319-39214-1

**Chapter 7**

**Provisional chapter**

**Interaction of Solitons with the Electromagnetic Field**

Solitary waves and solitons are briefly discussed in their historical context, especially regarding the motivation to use solitons for particle models in the nonlinear Klein-Gordon (NKG) equation. Conservation equations for charge, energy, and momentum follow from Noether's theorem in the usual way, and the NKG equation can be coupled to Maxwell's equations in the standard, gauge invariant manner. A recently proposed model is summarized in which two NKG equations are coupled to the electromagnetic field. In that model, solitons mimic the dynamical behavior of electrons and protons. A new result is then presented that follows from that model. Although the model is purely classical, it turns out that the arc spectrum of hydrogen is emitted into the electromagnetic field when small oscillations of one Klein-Gordon equation occur in the vicinity of a proton-like soliton. It is perhaps unexpected that a purely classical model can exhibit behavior suggestive of a phenomenon that is generally presumed to occur only in a quantum-mechanical context. Because of the way in which that result occurs, however, it is not clear whether there is any possible relevance to the actual physical phenomenon. **Keywords:** nonlinear systems and models, solitons, nonlinear Klein-Gordon-Maxwell

equations, particle models, hydrogen spectrum, nonlinear field theories

**Interaction of Solitons with the Electromagnetic Field**

DOI: 10.5772/intechopen.71215

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

<sup>∂</sup>*t*<sup>2</sup> <sup>−</sup> <sup>∇</sup><sup>2</sup> *<sup>ψ</sup>* <sup>+</sup> *<sup>ψ</sup>* <sup>=</sup> <sup>0</sup> (1)

and reproduction in any medium, provided the original work is properly cited.

When a term that acts as a restoring force is added to the usual linear wave equation, the equation

is obtained. Schweber ([1], p. 54) points out "When Schrödinger wrote down the nonrelativistic equation now bearing his name, he also formulated the corresponding relativistic

**in Classical Nonlinear Field Models**

**in Classical Nonlinear Field Models**

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.71215

Jon C. Luke

**Abstract**

**1. Introduction**