# Past and present trends in the development of the pattern-formation theory:
domain walls and quasicrystals

Boris A. Malomed1.2 1Department of Physical Electronics, School of Electrical
Engineering, Faculty of Engineering, and Center for Light-Matter Interaction,
Tel Aviv University, P.O. Box 39040 Tel Aviv, Israel
2Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica,
Chile

###### Abstract

A condensed review is presented for two basic topics in the theory of pattern
formation in nonlinear dissipative media: (i) domain walls (DWs, alias grain
boundaries), which appear as transient layers between different states
occupying semi-infinite regions, and (ii) two- and three-dimensional (2D and
3D) quasiperiodic (QP) patterns, which are built as superposition of plane-
wave modes with incommensurate spatial periodicities. These topics are
selected for the present article, dedicated to the 70th birthday of Professor
M. I. Tribelsky, due to the impact made on them by papers published by Prof.
Tribelsky and his coauthors. Although some findings revealed in those works
may now seem as “old” ones, they keep their significance as fundamentally
important results in the the theory of nonlinear DW and QP patterns. Adding to
the findings revealed in the original works by M. I. Tribelsky et al., the
present article also reports several new analytical results, obtained as exact
solutions to systems of coupled real Ginzburg-Landau (GL) equations. These
are: a new solution for symmetric DWs in the bimodal system including linear
mixing between its components; a solution for a strongly asymmetric DWs in the
case when the diffusion (second-derivative) term is present only in one GL
equation; a solution for a system of three real GL equations, for the
symmetric DW with a trapped bright soliton in the third component; and an
exact solution for DWs between counter-propagating waves governed by the GL
equations with group-velocity terms. The significance to the “old” and new
results collected in this review is enhanced by the fact that the systems of
coupled equations for two- and multicomponent order parameters, addressed in
the article, apply equally well to modeling thermal convection, multimode
light propagation in nonlinear optics, and binary Bose-Einstein condensates.

Dedicated to the celebration of the 70th birthday of Professor Mikhail
Isaakovich Tribelsky

##  I Introduction

###  I.1 The objective of this article

Apart from his fundamental contributions to optics, especially to the theory
of the nonresonant light-matter interaction [1, 2] and light scattering by
small particles [2]-[7], an essential topic in the works of Mikhail Tribelsky
[8] has been the theory of pattern formation in nonlinear dissipative media.
In particular, two important subjects considered in his publications are
domain walls (DWs, alias grain boundaries), i.e., stationary stripes
separating two domains which are filled by different stable patterns, and
quasiperiodic (QP) patterns, alias dissipative two-dimensional (2D)
quasicrystals. It is relevant to mention that the fundamental papers of Prof.
Tribelsky on the former and latter topics, viz., Refs. [9] and [10], are,
respectively, his second and sixth best-cited publications, according to the
data provided by Web of Science. The objective of this article is to produce a
condensed review of basic results reported in those old but still significant
works, and outline directions of subsequent work which was initiated by
results reported in them. The article also includes some new exact analytical
results for DWs which offer a natural extension of the analysis initiated in
Ref. [9] (in a detailed form, the new results will be reported elsewhere
[11]). The presentation given in the article has a personal flavor, due to the
fact that the present author was a Mikhail’s collaborator in projects which
had produced the above-mentioned original publications.

In addition Ref. [9], it is relevant to mentioned a still earlier paper [12],
where we addressed a well-known model equation, which is usually called the
real Ginzburg-Landau (GL) equation (the name originates from the
phenomenological theory of superconductivity elaborated by Ginzburg and Landau
70 years ago [13]). The usual scaled form of the real GL equation is

$\frac{\partial u}{\partial t}=u+\frac{\partial^{2}u}{\partial
x^{2}}-|u|^{2}u.$ (1)

Actually, the order parameter $u(x,t)$ governed by Eq. (1) is a complex
function, while the equation is called “real” because its coefficients are
real (therefore, by means of scaling, all coefficients in Eq. (1) are set to
be $\pm 1$). The first, second, and third terms on the right-hand side of Eq.
(1) represent, respectively, the linear gain, diffusion/viscosity (dispersive
linear loss), and nonlinear loss. The real GL equation is a universal model
for many nonlinear dissipative media, such as the Rayleigh-Bénard convective
instability in a shallow layer of a liquid heated from below [14] and
instability of a plane laser evaporation front [1].

Note that Eq. (1) may be represented in the gradient form,

$\frac{\partial u}{\partial t}=-\frac{\delta L}{\delta u^{\ast}},$ (2)

where $\delta/\delta u^{\ast}$ stands for the functional (Freché) derivative,
and

$L=\int_{-\infty}^{+\infty}\left(-|u|^{2}+\left|\frac{\partial u}{\partial
x}\right|^{2}+\frac{1}{2}|u|^{4}\right)dx$ (3)

is a real Lyapunov functional. A consequence of the gradient representation is
that $L$ may only decrease or stay constant in the course of the evolution,
$dL/dt\leq 0$. This fact simplifies the dynamics of the real GL equation.

This equation gives rise to a family of simple stationary plane-wave (PW)
solutions,

$u(x)=\sqrt{1-k^{2}}\exp\left(ikx\right),$ (4)

where real wavenumber $k$ takes values in the existence band,

$-1<k<+1.$ (5)

A nontrivial issue is stability of the PW solutions against small
perturbations. It can be naturally addressed by rewriting Eq. (1) in the
Madelung form (sometimes called the hydrodynamic representation), substituting

$u\left(x,t\right)=A\left(x,t\right)\exp\left(i\phi\left(x,t\right)\right),$
(6)

where $A$ and $\phi$ are real amplitude and phase. The substitution splits Eq.
(1) into a pair of real equations:

$\displaystyle\frac{\partial A}{\partial t}=A+\frac{\partial^{2}A}{\partial
x^{2}}-A\left(\frac{\partial\phi}{\partial x}\right)^{2}-A^{3},$ (7)
$\displaystyle A\frac{\partial\phi}{\partial
t}=A\frac{\partial^{2}\phi}{\partial x}+2\frac{\partial A}{\partial
x}\frac{\partial\phi}{\partial x}.$ (8)

In terms of this system, the PW solution (4) is written as

$A=\sqrt{1-k^{2}},\phi(x)=kx.$ (9)

In paper [12] (incidentally, it is the ninth best-cited publication of M. I.
Tribelsky), the stability of solution (9) was explored by means of
linearization of Eqs. (7) and (8) against small perturbations of the amplitude
and phase. We had thus found that the stability region in the existence band
(5) is

$-1/\sqrt{3}\leq k\leq+1/\sqrt{3}.$ (10)

In this region, the squared amplitude of the PW solution, $A^{2}(k)$, exceeds
$2/3$ of its maximum value, $A_{\max}^{2}\equiv 1$, which corresponds to
$k=0$:

$A^{2}\equiv 1-k^{2}\geq 2/3.$ (11)

At that time, we were not aware of the fact that this result, in the form of
Eq. (10) was established much earlier [15] by Wiktor Eckhaus [16] It is now
commonly known as the Eckhaus stability criterion (ESC). Later, we had learned
that some other people entering this research area had also independently
rediscovered the ESC. This fact had suggested our coauthor in Refs. [9] and
[10], Prof. Alexander Nepomnyashchy, to formulate a Nepomnyashchy criterion: a
necessary condition for successful work on the pattern-formation theory is the
ability of the researcher to re-derive the ESC from the scratch.

###  I.2 Complex Ginzburg-Landau equations: the formulation, plane waves, and
dissipative solitons

Before proceeding to the discussion of particular topics included in this
article, it is relevant to briefly recapitulate the main principles concerning
complex GL equations, as a class of fundamental models underlying the theory
of pattern formation under the combined action of linear gain and loss
(including the diffusion/viscosity), linear wave dispersion, nonlinear loss,
and nonlinear dispersion. In the case of the cubic nonlinearity, the generic
form of this equation is [17, 18]

$\frac{\partial u}{\partial
t}=gu+\left(a+ib\right)\frac{\partial^{2}u}{\partial
x^{2}}-\left(d+ic\right)|u|^{2}u,$ (12)

cf. its counterpart (1) with real coefficients. Here, constants $g>0$, $a\geq
0$, and $d>0$ represents, severally, the linear gain, diffusion coefficient,
and nonlinear loss. Further, coefficients $b$ and $c$, which may have any
sign, control the linear and nonlinear dispersion, respectively. Coefficient
$g$ in Eq. (CCGL) may include an imaginary part too, but such a frequency term
can be trivially removed by a transformation, $u(x,t)\equiv
u(x,t)\exp\left[i\mathrm{Im}(g)t\right]$.

By means of obvious rescaling of $t$, $x$, and $u$, one can fix three
coefficients in Eq. (12):

$g=a=d=1,$ (13)

unless the equation does not include the diffusion term, in which case $a=0$
is set. Equation (12) is written in the 1D form, while its multidimensional
version is obtained replacing $\partial^{2}u/\partial x^{2}$ by the Laplacian,
$\nabla^{2}u$.

Unlike Eq. (1), the complex GL equation (12) does not admit a gradient
representation (see Eq. (2)). In the case of relatively small real parts of
the coefficients, i.e., $a\ll|b|$, $d\ll|c|$, Eq. (12) may be treated as a
perturbed version of the nonlinear Schrödinger (NLS) equation. Methods of the
perturbation theory for NLS equations had been developed in detail long ago
[19].

The ubiquity of the complex GL equations is stressed by the title of the major
review by Aranson and Kramer [17], “The world of the complex Ginzburg-Landau
equation” – indeed, the great number of particular forms of such equations,
their various realizations and applications, and the great number of
solutions, obtained by means of numerical and approximate analytical methods,
form a “world” in itself. As concerns applications, complex GL equations
emerge not only in areas, such as optics of laser cavities [20, 21, 22, 23],
where they can be directly derived as basic physical models, with $u(x,t)$
being a slowly-varying amplitude of the optical field, but also in many other
areas of physics (hydrodynamics, electron-hole plasmas in semiconductors and
gas-discharge plasmas, chemical waves, etc.). In many cases, underlying
systems of basic equations are complicated, but complex GL equations may be
derived from them as asymptotic equations for long-scale small-amplitude (but,
nevertheless, essentially nonlinear) excitations [24, 25, 26]. In some cases,
equations of the complex-GL type may also be quite useful as phenomenological
models [17, 18].

While DW states are supported by a finite-amplitude PW background, it is
relevant to mention that complex GL equations may give rise to localized
states (dissipative solitons [29]-[35]). In particular, Eq. (12) admits an
exact solution,

$u=A\,\left[\cosh\left(\kappa
x\right)\right]^{-\left(1+i\mu\right)}\exp\left(-i\omega t\right),$ (14)

with a single set of parameter values, $A$, $\kappa$, $\mu$, and $\omega$,
given by cumbersome expressions [36, 37]. If the complex GL equation reduces
to a perturbed NLS equation, the dissipative soliton (14) can be obtained from
the NLS soliton by means of the perturbation theory, under condition $bc<0$
(otherwise, the underlying NLS equation does not have bright-soliton
solutions). However, solution (14) is always unstable, as the linear gain in
Eq. (12), represented by $g>0$, makes the zero background around the soliton
unstable.

Dissipative solitons of this type may be effectively stabilized, in a
nonstationary form, in a model including time-periodic alternation of linear
gain and loss, which implies replacing the constant coefficient $g$ in Eq.
(12) by function $g(t)$ periodically changing between positive and negative
values; in particular, it may be taken as a periodic array of amplification
pulses on top of a constant lossy background,

$g(t)=G\sum_{n=-\infty}^{+\infty}\delta\left(t-\tau n\right)-g_{0},$ (15)

with $G>0$ and $g_{0}>0$, $\tau$ being amplification period [27]. Another
option for the stabilization is the use of the dispersion management, i.e.,
replacing constant dispersion coefficient $b$ in Eq. (12) by function $b(t)$
which periodically jumps between positive and negative values, cf. Eq. (15)
[28].

The fact that the dissipative soliton (14) may be considered as an extension
of bright NLS solitons suggests that Eq. (12) may also support a solution
resembling the dark soliton of the NLS equation with the self-defocusing
nonlinearity. Indeed, such solutions were found buy Nozaki and Bekki in the
form of “holes” [38]. Although the holes, as well as DWs, are supported by a
stable PW background, they are completely different states, as DWs separate
_different_ PWs (see below), while the hole is built into a single PW.

A more sophisticated version of the complex GL equation admits the existence
of _stable_ stationary dissipative solitons, if the zero solution is stable,
i.e., the linear term must be lossy, corresponding to $g<0$ in Eq. (12). In
this case, it is necessary to include the cubic gain and quintic loss (the
latter term prevents the blowup). Thus, one arrives at the complex GL equation
with the cubic-quintic nonlinearity, which was first introduced by
Petviashvili and Sergeev [47] (actually, as a 2D equation) in the form of

$\frac{\partial u}{\partial
t}=gu+\left(a+ib\right)\frac{\partial^{2}u}{\partial
x^{2}}-\left(d+ic\right)|u|^{2}u-\left(f+ih\right)|u|^{4}u,$ (16)

with $g<0$, $a\geq 0$, $d<0$, and $f>0$, cf. Eq. (12).

It follows from Eq. (16) that the interplay of the gain and loss terms in Eq.
(16) allows generation of nonzero states under the condition that the cubic-
gain strength exceeds a minimum value necessary to compensate the effect of
the loss:

$|d|>\left(|d|\right)_{\min}=2\sqrt{|g|f}.$ (17)

Further, using the rescaling freedom, one can normalize Eq. (16) by setting

$-g=a=-d\equiv 1,$ (18)

cf. Eq. (13) in the case of Eq. (12). Then, condition (17) amounts to $f<1/4$.

Stable dissipative solitons as solutions of Eq. (16), in the case when they
may be considered as a perturbation of the NLS solitons, were first predicted
in Ref. [29], and later rediscovered in Ref. [48]. Then, stable dissipative-
soliton solutions of Eq. (16) were found in the opposite limit, when the
dispersive terms in this equation may be treated as small perturbations. In
this case, the dissipative solitons are broad (nearly flat) states, bounded by
sharp edges in the form of a kink and antikink [49, 50, 51].

The complex GL equation (16), subject to normalization (18), generates a
family of PW solutions, where the wavenumber takes values in the same interval
(5) as above:

$\psi=\sqrt{1-k^{2}}\exp\left(ikx-i\omega t\right),\,\omega=c+(b-c)k^{2},$
(19)

cf. stationary solutions (4) of the real GL equation. The stability of these
flat states against long-wave perturbations can be investigated analytically,
leading to a generalization of the ESC (cf. Eq. (10)):

$k^{2}\leq\left(1+bc\right)/\left(3+2c^{2}+bc\right)$ (20)

[52]. The full stability of solutions (19) was investigated in a numerical
form [17, 18]. Note that condition (20) cannot hold unless the dispersion
coefficients in Eq. (16), normalized as per Eq. (18), satisfy the Benjamin-
Feir-Newell (BFN) condition,

$1+bc>0.$ (21)

If this condition does not hold, unstable PWs develop phase turbulence, with
$|\psi|$ staying roughly constant, while the phase of the complex order
parameter, $\phi(x,t)\equiv\arg\left\\{u(x,t)\right\\}$, demonstrates
spatiotemporal chaos. Just below the BFN instability threshold, i.e., at
$0<-\left(1+bc\right)\ll 1$ (see Eq. (21)), the chaotic evolution of the phase
gradient $p\equiv\phi_{x}$ obeys the Kuramoto-Sivashinsky equation [53, 54],
whose scaled form is

$p_{t}+p_{xx}+p_{xxxx}+pp_{x}=0.$ (22)

Deeper into the region of $1+bc<0$, the instability creates defects of the
wave field, at which $|u\left(x,t\right)|=0$, and eventually leads to the
onset of defect turbulence [17, 18]. Further evolution may lead to emergence
of regularly arranged train-shaped patterns in the turbulent states [55].

###  I.3 The structure of the article

The rest of the paper is divided into main Sections II and III. The former one
addresses the concept of DWs and its further development, following Ref. [9].
The DWs considered in that work were constructed as solutions of a system of
two nonlinearly-coupled real GL equations, which model the interaction of two
families of simplest roll patterns (quasi-1D spatially periodic structures) in
the Rayleigh-Bénard convection. This setup is controlled by the
overcriticality,

$\varepsilon=\left(\mathrm{Ra}-\mathrm{Ra}_{\mathrm{crit}}\right)/\mathrm{Ra}_{\mathrm{crit}}~{},$
(23)

where $\mathrm{Ra}$ is the Rayleigh number, and $\mathrm{Ra}_{\mathrm{crit}}$
is its critical value at the threshold of the instability of the fluid layer
heated from below. DWs in convection patterns were predicted as linear defects
(“grain boundaries”) [14, 56, 57], and were directly observed in experiments,
both as DWs proper and more complex structures, formed by junctions of DWs.
Typical examples of the experimentally observed patterns, borrowed from Ref.
[58], are presented in Fig. 1.

Figure 1: (a) An experimentally observed pattern of rolls in the Rayleigh-
Bénard convection, which demonstrates a junction of domain walls (grain
boundaries). The pattern corresponds to overcriticality $\varepsilon=1$, see
Eq. (23). (b) Similar patterns observed at $\varepsilon=1.8$ (left) and $2$
(right). Reprinted from Ref. [58].

It is relevant to stress that the concept of grain boundaries is known, in a
great variety of different realizations, as a very general one in condensed-
matter physics [59, 60, 61, 62, 63, 64]. In most cases, the nature of such
objects is different from that in thermal convection and other nonlinear
dissipative media, Nevertheless, the phenomenology of the grain boundaries in
completely different physical systems has many common features.

The DW states were constructed in Ref. [9] as solutions of two coupled real GL
equations for amplitudes of PWs connected by the DW. In a particular case,
such a solution for a symmetric DW is available in an exact analytical form,
see Eqs. (50) and (51) below. It is also demonstrated that the symmetric DW
may play the role of a potential well which traps an additional small-
amplitude component, in the form of a bright soliton, thus making the
structure of the DW more complex, as shown below by Eqs. (72)-(75) and Fig. 5.
Further, a newly derived extension of the exact solution in included, for the
case when the symmetrically coupled real GL equations include linear-mixing
terms (see Eq. (54) below), and a new exact solution for a strongly asymmetric
DW, in the case when only one real GL equation includes the diffusion term
(second derivative). This solution is given below by Eqs. (64)-(68) and Fig.
4.

At the level of stationary solutions, the same coupled equations which model
the grain boundaries in thermal convection predict DWs in optics, as
boundaries between spatial or temporal domains occupied by PWs representing
different polarizations or different carrier frequencies of light [65]. These
equations also produce DW states in binary Bose-Einstein condensates (BECs)
composed of immiscible components [66].

Still earlier, approximate solutions similar to DWs were constructed in the
framework of a single complex GL equation [67]. Such solutions represent
stationary sources of stable PWs with wavenumbers $\pm k$ (see Eq. (19)),
emitted in opposite directions (while the above-mentioned “holes” [38] are
sinks absorbing colliding PWs with opposite wavenumbers). A special case
corresponds to the complex GL equation (12) without the diffusion term, i.e.,
with $a=0$. In that case, DWs may be approximately reduced to shock waves
governed by an effective Burgers equation for a local wavenumber [39]. These
results are also included, in a brief form, in Section II. As an extension of
the topic, this section also addresses DWs between semi-infinite domains
filled by counterpropagating traveling waves (this is possible, in particular,
in thermal convection in a layer of a binary fluid heated from below [68, 69,
70, 71]). Furthermore, Section II includes a newly found exact solution for
the DW between traveling waves, produced by a system of coupled real GL
equations that include group-velocity terms (see Eqs. (85)-(90) below).

Section III summarizes some theoretical results for QP patterns in 2D and 3D
nonlinear dissipative media, the study of which was initiated in Ref. [10]. In
particular, included are findings for stable QP states produced by
combinations of four spatial modes in a laser cavity with different 3D wave
vectors [82]. Another possibility to produce a spatially confined four-mode
(eight-fold) QP structure, briefly mentioned in Section III, is offered by the
overlap of two square-shaped (two-mode) patterns, under the angle of
$45^{\text{o}}$, in a transient layer between the patterns [83]. This
possibility is a combination of the two main topics considered in this
article, viz., DWs and QP patterns.

The article is completed by Section IV, which summarizes basic results and
briefly outlines new possibilities in this area.

##  II DW (domain-wall) patterns

###  II.1 The source pattern generated by the single complex GL equation

####  II.1.1 The generic case

To produce approximate solutions to Eq. (12), it is convenient to rewrite it
in the Madelung form (6), which yields the following system of equations for
real amplitude $A$ and phase $\phi$:

$\displaystyle\frac{\partial A}{\partial
t}=A-A^{3}+\frac{\partial^{2}A}{\partial
x^{2}}-A\left(\frac{\partial\phi}{\partial x}\right)^{2}-2b\frac{\partial
A}{\partial x}\frac{\partial\phi}{\partial
x}-bA\frac{\partial^{2}\phi}{\partial x^{2}},$ (24) $\displaystyle
A\frac{\partial\phi}{\partial t}=2\frac{\partial A}{\partial
x}\frac{\partial\phi}{\partial x}+A\frac{\partial^{2}\phi}{\partial
x^{2}}-cA^{3}+b\frac{\partial^{2}A}{\partial
x^{2}}-bA\left(\frac{\partial\phi}{\partial x}\right)^{2}$ (25)

(recall coefficients of Eq. (12) are subject to normalization conditions
(13)). As shown in Ref. [67], a stationary solution of the DW type, which
represents a source of PWs emitted in the directions of
$x\rightarrow\pm\infty$, can be looked for assuming that the dispersion
coefficients $b$ and $a$ are small, and the local amplitude, $A(x)$, and
wavenumber, $p(x)\equiv\partial\phi/\partial x$, are slowly varying functions
of $x$ (the “nonlinear geometric-optics approximation”, alias the “eikonal
approximation”). In the lowest order, all derivatives and dispersion terms may
be neglected in Eq. (24), reducing it merely to $A^{2}\approx 1-p^{2}$, cf.
Eq. (4). Next, this approximation is substituted in Eq. (25), with the phase
taken as

$\phi(x,t)=-\left(c+(b-c)k^{2}\right)t+\int p(x)dx,$ (26)

where it is assumed that the asymptotic values of the wavenumber are

$p\left(x\rightarrow\pm\infty\right)=\pm k$ (27)

(hence the frequency in expression (26) is the same as in Eq. (19)). Keeping
the lowest-order small terms with respect to the small dispersive coefficients
and small derivative $dp/dx$ of the slowly varying local wavenumber leads to
the following approximate equation:

$\frac{1-3p^{2}}{1-p^{2}}\frac{dp}{dx}=\left(c-b\right)\left(k^{2}-p^{2}\right).$
(28)

The DW solution to Eq. (28) can be obtained in an implicit form, which yields
$x$ as a function of $p$, satisfying the boundary conditions (27):

$2k\ln\frac{1-p}{1+p}+\left(1-3k^{2}\right)\ln\frac{k-p}{k+p}=2k\left(b-c\right)\left(1-k^{2}\right)x.$
(29)

The solution can be easily cast in an explicit form under condition $k^{2}\ll
1$:

$p(x)\approx k\tanh\left[\left(c-b\right)kx\right].$ (30)

This form clearly demonstrates that the DW may be indeed construed as an
emitter of waves from the center, where $p(x=0)=0$, to
$x\rightarrow\pm\infty$, in agreement with Eq. (28). The explicit solutions,
as well as the implicit ones (29), constitute a family parameterized by free
constant $k$.

In the real GL equation, with $b=c=0$, as well as in the case when the linear
and nonlinear dispersions exactly cancel each other, $b=c$, Eq. (28) cannot
produce a stationary DW solution. As shown in Ref. [67], in that case initial
configurations in the form resembling expression (30), i.e., a step-shaped
profile of the local wavenumber, give rise to nonstationary solutions, which
may be approximated by means of characteristics and caustics of a quasi-linear
evolution equation for $p(x,t)$.

####  II.1.2 Domain walls as shock waves in the diffusion-free complex GL
equation

The consideration of DWs should be performed differently in the special case
of the complex GL equation (12) with $a=0$, which does not include the
diffusion term. Taking into regard normalization (13), the respective equation
takes the form of

$\frac{\partial u}{\partial t}=u+ib\frac{\partial^{2}u}{\partial
x^{2}}-\left(1+ic\right)|u|^{2}u$ (31)

(in fact, one can additionally rescale coordinate $x$ here, to set $b=\pm 1$).
This form of the equation admits free motion of various modes [39, 30]. In
this case, the Madelung substitution (6) leads, instead of the amplitude-phase
equations (24) and (25), to ones

$\displaystyle\frac{\partial A}{\partial t}=A-A^{3}-2b\frac{\partial
A}{\partial x}\frac{\partial\phi}{\partial
x}-bA\frac{\partial^{2}\phi}{\partial x^{2}},$ (32) $\displaystyle
A\frac{\partial\phi}{\partial t}=-cA^{3}+b\frac{\partial^{2}A}{\partial
x^{2}}-bA\left(\frac{\partial\phi}{\partial x}\right)^{2}.$ (33)

Further, the lowest approximation of the nonlinear geometric optics, applied
to Eq. (32), yields

$A^{2}\approx 1-b\frac{\partial p}{\partial x}.$ (34)

The substitution of this in Eq. (33) leads, after simple manipulations
(including the division by $A$ and differentiation with respect to $x$, in
order to replace $\partial\phi/\partial t$ by $\partial p/\partial t$), to the
Burgers equation [39],

$\frac{\partial p}{\partial t}=bc\frac{\partial^{2}p}{\partial
x^{2}}-2bp\frac{\partial p}{\partial x}.$ (35)

The usual shock-wave solutions of Eq. (35) give rise to a family of DWs with
two independent parameters, viz., wall thickness $\xi>0$ and speed $s$, which
may be positive or negative:

$p(x,t)=\frac{s}{2b}-\frac{c}{\xi}\tanh\left(\frac{x-st}{\xi}\right).$ (36)

The appearance of the second free parameter, $s$, in this solution corresponds
to the above-mentioned fact that Eq. (31) admits free motion of patterns
produced by this variant of the complex GL equation.

###  II.2 DWs in systems of real coupled GL equations, as per Ref. [9], and
additional analytical findings

####  II.2.1 The setting

The starting point of the analysis developed in Ref. [9] was a general
expression for the distribution of the complex order parameter in the 2D
system (e.g., the amplitude of the convective flow):

$U(x,y;t)=\sum_{l=1}^{N}u_{l}(x,y;t)\exp\left(i\mathbf{n}_{l}\mathbf{\cdot
R}\right),$ (37)

where $\mathbf{R=}\left(x,y\right)$. Equation (37) implies that the order-
parameter field is a superposition of $N$ plane-wave modes (often called
rolls, in the context of the convection theory) with wave vectors
$\mathbf{n}_{l}$, and $u_{l}\left(x,y\right)$ are slowly varying amplitudes of
these modes. Stationary states produced by the real GL equations may be looked
for in the real form too,

$u_{l}(x,y)\equiv r_{l}\left(x,y\right),\arg\left(u_{l}\right)=0.$ (38)

as the evolution of phases $\arg\left(u_{l}\right)$ is trivial in this case.

It is relevant to mention that patterns similar to the rolls (known under the
same name) are produced by the Lugiato-Lefever (LL) equation and its
varieties. The basic LL equation may be considered as the NLS equation for
amplitude $u\left(x,t\right)$ of the optical field in a laser cavity, which
includes the linear-loss coefficient, $\gamma>0$, a real cavity-mismatch
parameter, $\theta\gtrless 0$, and a constant pump field, $u_{0}$ [40]:

$\frac{\partial u}{\partial t}=-\gamma
u+u_{0}+i\left(|u|^{2}-\theta\right)u+i\frac{\partial^{2}u}{\partial x^{2}}.$
(39)

Roll patterns were studied in details in various forms of LL models [41, 42,
43]. DWs also occur in these systems [44, 45].

As is illustrated by Fig. 2(a), the simplest possibility of the realization of
patterns represented by Eq. (37) is the superposition of $N=2$ modes, each one
filling, essentially, a half-plane bounded by the DW. In this case, real
amplitudes $r_{1.2}$ are slowly varying functions of only one coordinate, $x$,
directed perpendicular to the DW. The respective boundary conditions (b.c.)
are

$\displaystyle
r_{1}\left(x\rightarrow-\infty\right)=r_{2}\left(x\rightarrow+\infty\right)=\mathrm{const}\neq
0,$ $\displaystyle
r_{1}\left(x\rightarrow+\infty\right)=r_{2}\left(x\rightarrow-\infty\right)=0.$
(40)

The scaled form of stationary (time-independent) coupled real GL equations for
slowly varying amplitudes $r_{1}(x)$ and $r_{2}(x)$, corresponding to the
bimodal DW configuration defined as per Fig. 2(a) and Eq. (40), are [9] (see
also Refs. [14] and [56])

$\displaystyle
D_{1}\frac{d^{2}r_{1}}{dx^{2}}+r_{1}\left(1-r_{1}^{2}-Gr_{2}^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ (41) $\displaystyle
D_{2}\frac{d^{2}r_{2}}{dx^{2}}+r_{2}\left(1-r_{2}^{2}-Gr_{1}^{2}\right)$
$\displaystyle=$ $\displaystyle 0.$ (42)

Here effective diffusion coefficients are

$D_{1.2}=\cos^{2}\theta_{1,2}$ (43)

(see Fig. 2(a)), and $G>0$ is an effective coefficient of the cross-
interaction between different plane waves, while the self-interaction
coefficient is scaled to be $1$.

Figure 2: The scheme of the formation of the DW (domain wall) between 2D
patterns in the Rayleigh-Bénard convection and similar settings. (a) The DW
between plane waves (rolls) with wave vectors oriented under angles
$\theta_{1}$ and $\theta_{2}$ with respect to the $x$ axis, see Eq. (37). The
respective amplitudes $r_{1,2}(x)$ satisfy Eqs. (41) and (42) and are subject
to b.c. (40). The position of the DW is shown by the vertical dashed line. An
example of the DW profile is displayed below in Fig. 3(a). (b) The same as in
(a), for the DW between hexagons (the triple-mode pattern) and single-mode
rolls. The figure is reprinted from Ref. [9].

The symmetric configuration corresponds to Fig. 2(a) with

$\theta_{1}=-\theta_{2},$ (44)

which implies $D_{1}=D_{2}\equiv D$, according to Eq. (43). Naturally, the
symmetric case plays an important role in the analysis, as shown below.

It is relevant to mention that coupled equations (41) and (42) may be
considered as formal equations of motion for a mechanical system with two
degrees of freedom, while $x$ plays the role of formal time. This system keeps
a constant value of its (formal) Hamiltonian,

$h=\frac{1}{2}\sum_{j=1,2}\left[D_{j}\left(\frac{dr_{j}}{dx}\right)^{2}+r_{j}^{2}-\frac{1}{2}r_{j}^{4}\right]-\frac{G}{2}r_{1}^{2}r_{2}^{2}.$
(45)

DW solutions can be readily found as numerical solutions of coupled equations
(41) and (42), subject to b.c. (40). A characteristic example of the solution
is displayed in Fig. 3(a). In fact, the existence of the DWs in the framework
of Eqs. (41) and (42) may be understood as immiscibility of the modes whose
amplitudes are produced by these equations. The general condition for the
immiscibility, in the present notation, is well known since long ago:

$G>1,$ (46)

i.e., the strength of the mutual repulsion of the two components must exceed
the strength of their self-repulsion [46].

Figure 3: (a) A typical profile of the DW between different plane-wave (roll)
families. (b) The structure of the DW between the plane-wave and hexagonal
patterns (in this panel, $\mathbf{k}_{1,2,3}$ are identical to
$\mathbf{n}_{l}$ in Fig. (37)). The figure is reprinted from Ref. [9].

Hexagonal states in the Rayleigh-Bénard convection are produced by a
superposition of three plane waves, with angles $120^{\mathrm{o}}$ between
their wave vectors. Such patterns are stable if, in addition to the cubic
inter-mode interaction in Eqs. (41) and (42), the respective system of three
GL equations for local amplitudes $r_{1,2,3}(x)$ includes resonant quadratic
terms:

$D_{1}\frac{d^{2}r_{1}}{dx^{2}}+r_{1}\left[1-r_{1}^{2}-G\left(r_{2}^{2}+r_{3}^{2}\right)\right]+\nu
r_{2}r_{3}=0$ (47)

plus two complementing equations obtained from Eq. (47) by cyclic permutations
of subscripts $\left(1,2,3\right)$, where $\nu$ is a coefficient of the
resonant interaction. In the theory of the thermal convection, the quadratic
terms represent effects beyond the framework of the basic Boussinesq
approximation [93, 94]. Numerical solution of Eq. (47) produces DWs connecting
single-mode rolls and the hexagonal pattern [9], see an illustration in Fig.
2(b) and the corresponding pattern displayed in Fig. 3(b). It was also
demonstrated that DWs are possible between two bimodal (square-shaped)
patterns, each one composed of two plane waves with perpendicular
orientations. In that case, the DW appears as a boundary between two half-
planes filled by square patterns with different orientations [9, 83], see
further details below in Eqs. (125)-(128) and Fig. 9. In Ref. [84], a
spatially inhomogeneous model, in which the cross-interaction coefficient is a
function of the coordinate, $G=G(x)$, was introduced, making it possible to
construct stable DWs between the single- and bimodal patterns.

####  II.2.2 Basic analytical results from Ref. [9]

An analytically tractable case is the symmetric one, with $D_{1}=D_{2}\equiv
D$, and

$0<G-1\ll 1$ (48)

(recall that $G>1$ is a necessary condition for the existence of DWs). The
analysis makes it possible to reduce these coupled equations to an effective
sine-Gordon equation for a slowly varying inter-component phase $\chi(x)$, and
thus produce an approximate analytical DW solution with a large width of the
transient layer, $L\sim\left(G-1\right)^{-1/2}$ [9]:

$\left\\{\begin{array}[]{c}r_{1}(x)\\\
r_{2}(x)\end{array}\right\\}\approx\left\\{\begin{array}[]{c}\cos\chi(x)\\\
\sin\chi(x)\end{array}\right\\},\chi=\arctan\left(\exp\left(\sqrt{\frac{G-1}{D}}x\right)\right).$
(49)

In the particular case of

$G=3,$ (50)

the symmetric version of Eqs. (41) and (42) admits an _exact_ DW solution [9]:

$\left\\{\begin{array}[]{c}r_{1}(x)\\\
r_{2}(x)\end{array}\right\\}=\frac{1}{2}\left\\{\begin{array}[]{c}1-\tanh\left(x/\sqrt{2D}\right)\\\
1+\tanh\left(x/\sqrt{2D}\right)\end{array}\right\\},$ (51)

which is obviously compatible with b.c. (40).

####  II.2.3 New analytical results

##### Symmetric DWs

Precisely the same real time-independent equations as Eqs. (41) and (42)
appear in nonlinear optics as a stationary version of coupled NLS equations in
bimodal waveguides, with $r_{1}$ and $r_{2}$ being local amplitudes of
electromagnetic waves carrying different wavelengths or different
polarizations of light [85]. In the latter case, typical values of $G$ are
$2/3$ or $2$, for the linear or circular polarizations of the light,
respectively. Other values are possible too, in photonic-crystal waveguides
[86]. Similarly, the stationary real equations naturally appear as the time-
independent version of coupled Gross-Pitaevskii equations for mean-field wave
functions of binary BECs in ultracold atomic gases [87].

Thus, the same solutions as considered here may represent optical DWs [65], as
well as DWs separating two immiscible species in the BEC [66]. Further, the
coupled equations in optics and BEC models may also include linear mixing
between the interacting modes. In particular, this effect is produced by twist
applied to the bulk optical waveguide. A similar effect in binary BEC, viz.,
mutual inter-conversion of two atomic states, which form the binary BEC, may
be induced by the resonant radio-frequency field [88]. The respectively
modified symmetric system of Eqs. (41) and (42) is

$\displaystyle
D\frac{d^{2}r_{1}}{dx^{2}}+r_{1}\left(1-r_{1}^{2}-Gr_{2}^{2}\right)+\lambda
r_{2}$ $\displaystyle=$ $\displaystyle 0,$ (52) $\displaystyle
D\frac{d^{2}r_{2}}{dx^{2}}+r_{2}\left(1-r_{2}^{2}-Gr_{1}^{2}\right)+\lambda
r_{1}$ $\displaystyle=$ $\displaystyle 0,$ (53)

where real $\lambda$ is the linear-coupling coefficient. In fact, Eqs. (52)
and (53) apply to the Rayleigh-Bénard convection too, in the case when
periodic corrugation of the bottom of the convection cell, with amplitude
$\sim\lambda$ and wave vector $\mathbf{n}_{1}+\mathbf{n}_{2}$ (see Eq. (37)),
gives rise to the effect of the linear cross-gain. Actually, it is used in
many laser setups that are similar to thermal convection [89, 90].

The system of Eqs. (52) and (53) with $G=3$ admits an exact DW solution, which
is an extension of its counterpart (51):

$\left\\{\begin{array}[]{c}r_{1}(x)\\\
r_{2}(x)\end{array}\right\\}=\frac{1}{2}\left\\{\begin{array}[]{c}\sqrt{1+\lambda}-\sqrt{1-\lambda}\tanh\left(\sqrt{\frac{1-\lambda}{2D}}x\right)\\\
\sqrt{1+\lambda}+\sqrt{1-\lambda}\tanh\left(\sqrt{\frac{1-\lambda}{2D}}x\right)\end{array}\right\\}.$
(54)

Due to the action of the linear mixing, b.c. (40) are replaced by

$\displaystyle
r_{1}\left(x\rightarrow-\infty\right)=r_{2}\left(x\rightarrow+\infty\right)=\frac{1}{2}\left(\sqrt{1+\lambda}+\sqrt{1-\lambda}\right),$
$\displaystyle
r_{1}\left(x\rightarrow+\infty\right)=r_{2}\left(x\rightarrow-\infty\right)=\frac{1}{2}\left(\sqrt{1+\lambda}-\sqrt{1-\lambda}\right).$
(55)

The extended exact solution given by Eqs. (52)-(55) was not reported in
previous publications.

##### The effect of the confining potential

The above-mentioned realization of the coupled real GL equations in terms of
the binary BEC should include, in the general case, a trapping harmonic-
oscillator (HO) potential, which is normally used in the experiment [87]. The
accordingly modified system of Eqs. (52) and (53) is

$\displaystyle
D\frac{d^{2}r_{1}}{dx^{2}}+r_{1}\left(1-r_{1}^{2}-Gr_{2}^{2}\right)+\lambda
r_{2}$ $\displaystyle=$ $\displaystyle\frac{\aleph^{2}}{2}x^{2}r_{1},$ (56)
$\displaystyle
D\frac{d^{2}r_{2}}{dx^{2}}+r_{2}\left(1-r_{2}^{2}-Gr_{1}^{2}\right)+\lambda
r_{1}$ $\displaystyle=$ $\displaystyle\frac{\aleph^{2}}{2}x^{2}r_{2},$ (57)

where $\aleph^{2}$ is the strength of the OH potential. DW solutions of the
system of Eqs. (56) and (57) were addressed in Ref. [91]. A rigorous
mathematical framework for the analysis of such solutions in the absence of
the linear coupling ($\lambda=0$) was elaborated in Ref. [92].

If the HO trap is weak enough, viz., $\aleph^{2}\ll 4/\left(1-\lambda\right)$,
the DW trapped in the OH potential takes nearly constant values, close to
those in Eq. (55), in the region of

$2D/\left(1-\lambda\right)\ll x^{2}\ll 8D/\aleph^{2}.$ (58)

At $x^{2}\rightarrow\infty$, solutions generated by Eqs. (56) and (57) decay
similar to eigenfunctions of the HO potential in quantum mechanics, viz.,

$\displaystyle r_{1,2}$ $\displaystyle\approx$
$\displaystyle\varrho_{1,2}|x|^{\beta}\exp\left(-\frac{\aleph}{2\sqrt{2D}}x^{2}\right),$
(59) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{1+\lambda}{\sqrt{2D}\aleph}-\frac{1}{2},$ (60)

where $\varrho_{1,2}$ are constants. In the case of $\lambda=0$, the
asymptotic tails (59) follow the structure of solution (51), i.e.,
$\varrho_{1}\left(x\rightarrow+\infty\right)=\varrho_{2}\left(x\rightarrow-\infty\right)=0$
and
$\varrho_{1}\left(x\rightarrow-\infty\right)=\varrho_{2}\left(x\rightarrow+\infty\right)\neq
0$. On the other hand, the presence of the linear mixing, $\lambda\neq 0$,
makes the tail symmetric with respect to the two components,
$\varrho_{1}\left(|x|\rightarrow\infty\right)=\varrho_{2}\left(|x|\rightarrow\infty\right)\neq
0$. Note that $\beta=0$ in Eq. (60) with $\lambda=0$ is tantamount to the case
when values of $\aleph$ and $D$ in Eqs. (56) and (57) correspond to the ground
state of the HO potential.

##### Exact asymmetric DWs

Another possibility to add a new analytical solution for DWs appears in the
limit case of the extreme asymmetry in the system of Eqs. (41) and (42), which
corresponds to $D_{2}=0$ and $D_{1}\equiv D>0$, i.e., the DW between two roll
families one of which has the wave vector perpendicular to the $x$ axis, see
Eq. (43):

$\displaystyle
D\frac{d^{2}r_{1}}{dx^{2}}+r_{1}\left(1-r_{1}^{2}-Gr_{2}^{2}\right)=0,$ (61)
$\displaystyle r_{2}\left(1-r_{2}^{2}-Gr_{1}^{2}\right)=0.$ (62)

Actually, the form of Eq. (62), in which the second derivative drops out,
corresponds to the well-known Thomas-Fermi approximation (TF) in the BEC
theory. In the framework of the TF approximation, the kinetic-energy term in
the Gross-Pitaevskii equation is neglected, in comparison with larger ones,
representing a trapping potential and the (self-repulsive) nonlinearity [87].
In the present case, corresponding to $\theta_{2}=90^{\mathrm{o}}$, i.e.,
$D_{2}=0$ in Eq. (43), is not an approximation, but the exact special case. As
concerns the application of Eqs. (41) and (42) to BEC, with the kinetic-energy
coefficients $D_{1,2}=\hbar^{2}/\left(2m_{1,2}\right)$ in physical units,
where $m_{1,2}$ are atomic masses of the two components of the heteronuclear
binary condensate, Eqs. (61) and (62) correspond to a semi-TF approximation,
representing a mixture of light (small $m_{1}$) and heavy (large $m_{2}$)
atoms, e.g., a 7Li–87Rb diatomic gas [95].

Obviously, Eq. (62) yields two solutions, viz., either

$r_{2}=0,$ (63)

or one featuring to the quasi-TF relation,

$r_{2}^{2}(x)=1-Gr_{1}^{2}(x).$ (64)

In the former case, Eq. (61) with $r_{2}=0$ yields a solution in the form of
the usual dark soliton, while in the latter case, the substitution of
expression (64) in Eq. (61) produces a bright-soliton solution. These
solutions may be “dovetailed” at a stitch point,

$x=x_{0}\equiv-\sqrt{2D}\ln\left(\frac{\sqrt{G}+1}{\sqrt{G}-1}\right),$ (65)

which is defined by condition $r_{1}^{2}(x)=1/G$ (see Eq. (64). The global
form of the solution, which complies with b.c. (40), is

$r_{1}(x)=\left\\{\begin{array}[]{c}-\tanh\left(x/\sqrt{2D}\right),~{}\mathrm{at}~{}-\infty<x<x_{0},\\\
\sqrt{\frac{2}{G+1}}\mathrm{sech}\left[\sqrt{\frac{G-1}{D}}\left(x-\xi\right)\right],\mathrm{at}~{}x_{0}<x<+\infty,\end{array}\right.$
(66)
$r_{2}(x)=\left\\{\begin{array}[]{c}0,~{}\mathrm{at}~{}-\infty<x<x_{0},\\\
\sqrt{1-Gr_{1}^{2}(x)},\mathrm{at}~{}x_{0}<x<+\infty.\end{array}\right.$ (67)

Finally, the virtual center of bright-soliton segment of $r_{1}(x)$ is located
at

$x=\xi\equiv
x_{0}-\sqrt{\frac{D}{G-1}}\ln\left(\sqrt{\frac{2G}{G+1}}+\sqrt{\frac{G-1}{G+1}}\right)$
(68)

(actually, exact solution (66) makes use of the “tail” of the bright soliton
in the region of $x\geq x_{0}$, which does not include the central point,
$x=\xi$). The distance $x_{0}-\xi$, determined by Eq. (68), defines the
effective width of the strongly asymmetric DW. Note that, as seen in Eqs. (65)
and (66), this exact solution exists under the condition of $G>1$, which is
the above-mentioned immiscibility condition.

It is easy to check that expression (66) satisfies the continuity conditions
for $r_{1}(x)$ and $dr_{1}/dx$ at $x=x_{0}$, and expression (67) provides the
continuity of $r_{2}(x)$ at the same point. The continuity of $dr_{2}/dt$ at
$x=x_{0}$ is not required, as Eq. (62) does not include derivatives. A typical
example of the exact solution is displayed, for $D=1$ and $G=2$, in Fig. 4.

Figure 4: An example of the asymmetric DW, as given by Eqs. (65)-(68), for
$D=1$ and $G=2$ (here, $u_{1}$ and $u_{2}$ stand for $r_{1}$ and $r_{2}$ in
the analytical solution). Note that the coordinate of the stitch point is, in
this case, $x_{0}\approx-1.25$, as per Eq. (65), and the “virtual center” of
the bright-soliton segment of $u_{1}(x)$ is located at $\xi\approx-1.80$, as
per Eq. (68).

A remarkable fact is that, unlike the above-mentioned exact solutions (51) and
(54), which exist solely at $G=3$ (see Eq. (50)), the one given by Eqs.
(65)-(68) exists, as a generic one, for all values of $G>1$.

####  II.2.4 DW-bright-soliton complexes

##### An exact solution for the composite state

The DW formed by two immiscible PWs may serve as an effective potential for
trapping an additional PW mode. To address this possibility, it is relevant to
consider the symmetric configuration, with $D_{1}=D_{2}\equiv D$ (see Eq.
(43)), and wave vector $\mathbf{k}_{v}$ of the additional PW mode, $v(x)$,
directed along the bisectrix of the angle between the DW-forming wave vectors
$\mathbf{k}_{1}$ and $\mathbf{k}_{2}$, i.e., along axis $x$ (hence Eq. (43)
yields $D_{v}=1$). The corresponding system of three coupled stationary real
GL equations is

$\displaystyle
D\frac{d^{2}u_{1}}{dx^{2}}+u_{1}\left(1-u_{1}^{2}-Gu_{2}^{2}-gv^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ (69) $\displaystyle
D\frac{d^{2}u_{2}}{dx^{2}}+u_{2}\left(1-u_{2}^{2}-Gu_{1}^{2}-gv^{2}\right)$
$\displaystyle=$ $\displaystyle 0,$ (70)
$\frac{d^{2}v}{dx^{2}}+\left(1-v^{3}-g\left(u_{1}^{2}+u_{2}^{2}\right)\right)v=0,$
(71)

where $g>0$ is the constant of the nonlinear interaction between components
$u_{1,2}$ and $v$.

The system of Eqs. (69)-(71) admits the following exact solution, in the form
of the DW of components $u_{1,2}(x)$ coupled to a bright-soliton profile of
$v(x)$:

$\left\\{\begin{array}[]{c}u_{1}(x)\\\
u_{2}(x)\end{array}\right\\}=\frac{1}{2}\left\\{\begin{array}[]{c}1-\tanh\left(\sqrt{g-1}x\right)\\\
1+\tanh\left(\sqrt{g-1}x\right)\end{array}\right\\},$ (72)
$v(x)=\sqrt{2-\frac{3}{2}g}\mathrm{sech}\left(\sqrt{g-1}x\right).$ (73)

This solution is valid under the condition that coefficients $G$ and $D$ in
Eqs. (69) and (70) take the following particular values,

$G=3-8g+6g^{2},$ (74) $D=\frac{1}{2}\left(3g-1\right).$ (75)

As is follows from Eq. (73), this solution contains free parameter $g$, which
may vary in a narrow interval,

$1<g<4/3$ (76)

(see also Eq. (82) below). According to Eqs. (74) and (75), the interval (76)
corresponds to coefficients $G$ and $D$ varying in intervals

$1<G<3;~{}1<D<3/2.$ (77)

Thus, adding the $v$ component lifts the degeneracy of the exact DW solution
(51), which exists solely at $G=3$.

Recall that, in the model of convection patterns, $D$ cannot take values
$D>1$, which disagrees with Eq. (77). However, values $D>1$ are relevant for
systems of Gross-Pitaevskii equations for the heteronuclear three-component
BEC. In the latter case, $D$ is the ratio of atomic masses of the different
species which form the triple immiscible BEC. Similarly, $D$ is the ratio of
values of the normal group-velocity dispersion of copropagating waves in the
temporal-domain realization of the real GL equations in nonlinear fiber optics
[65]. In the latter case, values $D>1$ are relevant too.

An example of the DW-bright-soliton complex is displayed in Fig. 5 for
$g=7/6$, in which case Eqs. (74) and (75) yield $G=11/6$ and $D=5/4$
(according to Eqs. (74) and (43)).

Figure 5: An example of the exact solution for the DW-bright-soliton complex,
given by Eqs. (72) and (73), for $g=7/6,G=11/6,$ and $D=5/4$.

##### The bifurcation of the creation of the composite state in the general
case

If relation (74) is not imposed on the interaction coefficients $g$ and $G$,
the solution for the composite state cannot be found in an exact form.
Nevertheless, it is possible to identify bifurcation points at which component
$v$ with an infinitesimal amplitude appears. To this end, Eq. (71) should be
used in the form linearized with respect to $v$:

$\frac{d^{2}v}{dx^{2}}+\left\\{1-g\left[u_{1}^{2}(x)+u_{2}^{2}(x)\right]\right\\}v=0.$
(78)

This linear equation can be exactly solved for $u_{1}(x)=u_{2}(x)$ given by
expression (51), in the case of $G=3$, while parameters $D$ and $g$ may take
arbitrary values. Indeed, using the commonly known solution for the Pöschl-
Teller potential in quantum mechanics, it is easy to find that Eq. (78) with
the effective potential corresponding to expression (51) gives rise to
eigenmodes in the form of

$v(x)=\mathrm{const}\cdot\left[\mathrm{sech}\left(x/\sqrt{2D}\right)\right]^{\alpha},$
(79)

at a special value of the interaction coefficient, which identifies the
bifurcation producing the composite state:

$g_{\mathrm{bif}}=D^{-1}\left(1+2D\mp\sqrt{1+2D}\right),$ (80)

the respective value of power $\alpha$ in expression (79) being

$\alpha=\sqrt{2\left(1+D\mp\sqrt{1+2D}\right)}.$ (81)

The values given by Eqs. (80) and (81) with the top sign from $\mp$ correspond
to the bifurcation creating a fundamental composite state (the ground state,
in terms of the quantum-mechanical analog) at $g>g_{\mathrm{bif}}$, while the
bottom sign represents a higher-order bifurcation (alias the second excited
state, in the language of quantum mechanics; the first excited state, which is
not considered here, is a spatially odd mode). While it is obvious that the
fundamental bifurcation provides a stable composite state, it is plausible
that the ones produced by higher-order bifurcations are unstable.

Lastly, varying coefficient $D$ of the modes forming the underlying DW between
$D=0$ and $D=\infty$ (recall that the convection model corresponds to $D<1,$
while the realizations in BEC and optics admit $D>1$), Eq. (80) demonstrates
monotonous variation of the bifurcation point in interval

$g_{\mathrm{bif}}(D=0)\equiv 1<g_{\mathrm{bif}}<2\equiv
g_{\mathrm{bif}}\left(D\rightarrow\infty\right).$ (82)

It extends interval (76) in which exact composite states with a finite
amplitude were found above, see Eqs. (72)-(75).

###  II.3 Domain walls between traveling waves

####  II.3.1 The setting

An essential extension of the above results for DWs, produced by coupled
equations (41) and (42), was reported in Ref. [96], which addressed a system
of coupled GL equations for counter-propagating traveling waves, such as those
occurring in binary-fluid convection [68, 69, 70, 71]. The system is composed
of two equations of type (12) (subject to normalization (13)), coupled by
complex cubic terms with coefficients $G$ and $H$:

$\displaystyle\frac{\partial u_{1}}{\partial t}+s\frac{\partial
u_{1}}{\partial x}$ $\displaystyle=$ $\displaystyle
u_{1}+\left(1+ib\right)\frac{\partial^{2}u_{1}}{\partial
x^{2}}-\left(1+ic\right)|u_{1}|^{2}u_{1}-\left(G+iH\right)|u_{2}|^{2}u_{1},$
(83) $\displaystyle\frac{\partial u_{2}}{\partial t}-s\frac{\partial
u_{1}}{\partial x}$ $\displaystyle=$ $\displaystyle
u_{2}+\left(1+ib\right)\frac{\partial^{2}u_{2}}{\partial
x^{2}}-\left(1+ic\right)|u_{2}|^{2}u_{2}-\left(G+iH\right)|u_{1}|^{2}u_{2},$
(84)

where $-s$ and $+s$ are group velocities of the counter-propagating waves,
$u_{1}$ and $u_{2}$ (real coefficient $H$ represents the cross-phase
modulation (XPM), in terms of optics [85]). A natural approach to constructing
DW solutions of the system of Eqs. (83) and (84) is to use the lowest
approximation which neglects imaginary parts of coefficients in the equations,
but keeps the group-velocity terms. In this approximation, the order
parameters are real, $u_{1,2}(x)\equiv r_{1,2}(x)$, obeying the time-
independent version of Eqs. (83) and (84):

$\displaystyle+s\frac{dr_{1}}{dx}$ $\displaystyle=$
$\displaystyle\frac{d^{2}r_{1}}{dx^{2}}+r_{1}\left(1-r_{1}^{2}-Gr_{2}^{2}\right),$
(85) $\displaystyle-s\frac{dr_{2}}{dx}$ $\displaystyle=$
$\displaystyle\frac{d^{2}r_{2}}{dx^{2}}+r_{2}\left(1-r_{2}^{2}-Gr_{1}^{2}\right).$
(86)

Note that, unlike similar equations (41) and (42), Eqs. (85) and (86) cannot
be derived from a formal Hamiltonian, cf. Eq. (45).

####  II.3.2 A (new) exact analytical solution

To illustrate the structure of the DW state in this approximation, it is
relevant to produce a particular _exact solution_ of the system of equations
(85) and (86), cf. the above-mentioned solution (51):

$\left\\{\begin{array}[]{c}r_{1}(x)\\\
r_{2}(x)\end{array}\right\\}=\frac{1}{2}\left\\{\begin{array}[]{c}1-\tanh\left(\left(\sqrt{8+s^{2}}+s\right)(x/4\right)\\\
1+\tanh\left(\left(\sqrt{8+s^{2}}+s\right)(x/4\right)\end{array}\right\\}.$
(87)

This solution (which was not reported in earlier works) exists if the
following relation holds between the cross-interaction coefficient $G$ and
group velocity $v$:

$G-3=s\left(\sqrt{8+s^{2}}+s\right),$ (88)

or, inversely,

$s=\frac{G-3}{\sqrt{2\left(G+1\right)}}.$ (89)

cf. Eq. (50). It follows from the form of solution (87) and Eqs. (88) and (89)
that

$\mathrm{sgn}(s)=\mathrm{sgn}\left(G-3\right),$ (90)

i.e., the exact solution represents a sink of traveling waves ($s>0$) for
$G>3$, and a source ($s<0$) for $G<3$. Note that the solution of the latter
type exists even in the case of $G<1$, when the two components are miscible,
cf. Eq. (46). In this case, the mixing is prevented by the opposite group
velocities, which pull the components apart.

####  II.3.3 The sink or source coupled to a bright soliton in an additional
component

It is possible to consider a system including the counterpropagating traveling
waves coupled to an additional standing one. This is a natural counterpart of
the three-component system based on Eqs. (69), (70), and (71). The traveling
waves which can trap the additional standing one, $v(x)$, are described by the
following generalization of Eqs. (85) and (86):

$\displaystyle+s\frac{du_{1}}{dx}$ $\displaystyle=$
$\displaystyle\frac{d^{2}u_{1}}{dx^{2}}+u_{1}\left(1-u_{1}^{2}-Gu_{2}^{2}-gv^{2}\right),$
(91) $\displaystyle-s\frac{du_{2}}{dx}$ $\displaystyle=$
$\displaystyle\frac{d^{2}u_{2}}{dx^{2}}+u_{2}\left(1-u_{2}^{2}-Gu_{1}^{2}-gv^{2}\right),$
(92)

while the equation for the standing component is

$\frac{d^{2}v}{dx^{2}}+\left(1-v^{2}-g\left(u_{1}^{2}+u_{2}^{2}\right)\right)v=0,$
(93)

cf. Eq. (71). An exact solution of Eqs. (91)-(93) can be found for free
parameters $g$ and $s$:

$\displaystyle u_{1,2}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1\mp\tanh\left(\sqrt{g-1}x\right)\right),$ (94)
$\displaystyle v(x)$ $\displaystyle=$
$\displaystyle\sqrt{2-\frac{3}{2}g}\mathrm{sech}\left(\sqrt{g-1}x\right),$
(95) $\displaystyle G-3$ $\displaystyle=$ $\displaystyle
2g\left(3g-4\right)+4s\sqrt{g-1},$ (96) $\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{s}{2\sqrt{g-1}}+\frac{1}{2}\left(3g-1\right),$ (97)

cf. Eqs. (72)-(75). As it is seen from Eq. (96), the interaction with the
soliton-shaped standing wave shifts the boundary between the sink and source
of the traveling waves off the above-mentioned point, $G=3$.

##  III Two- and three-dimensional quasiperiodic patterns

Quasicrystals, as stable 3D ordered states of metallic alloys whose atomic
lattice is spatially quasiperiodic (QP), were discovered by D. Shechtman et
al. [72]. For this discovery, Shechtman was awarded with the Nobel Prize in
chemistry (2011). Then, a 2D quasicrystalline structure was also
experimentally demonstrated in alloys [73]. The work on this topic remains
very active in diverse branches of condensed-matter physics [74, 75, 76], as
well as in other physical systems which offer a natural realization of QP
patterns, such as photonics [77, 77, 79, 80] and phononics [81].

The objective of this section is to summarize results for stable 2D and 3D
patterns with the quasicrystalline structure that were predicted, as stable
non-equilibrium dynamical structures (rather than equilibrium states of
matter), in nonlinear dissipative media.

###  III.1 2D octagonal (eight-mode) and decagonal (ten-mode) quasicrystals

Following Ref. [10], generic 2D patterns of a real order parameter, such as
one representing the convection flow, are defined by means of complex
amplitudes $r_{l}$ of PWs which build them, cf. Eq. (37):

$U\left(x,y,t\right)=\sum_{l=1}^{2N}u_{l}(t)\exp\left(i\mathbf{n}_{l}\mathbf{\cdot
R}\right),$ (98)

where the set of $2N$ vectors $\mathbf{n}_{l}$ is a star with angles $\pi/N$
between adjacent ones. Note that the vectors satisfy the relation

$\mathbf{n}_{l+N}=-\mathbf{n}_{l},~{}\mathrm{for}~{}l=1,2,...,N.$ (99)

Amplitudes $u_{l}(t)$ are, generally speaking, complex variables,

$u_{l}(t)=A_{l}(t)\exp\left(i\varphi_{l}(t)\right)$ (100)

(cf. Eq. (6)), subject to constraint $u_{l+N}=u_{l}^{\ast}$, which, along with
Eq. (99), secures that the order-parameter distribution (98) is real.

For the lowest-order quasi-crystalline patterns, such as those corresponding
to $N=4$ (octagonal) and $N=5$ (decagonal) ones, the phase evolution is
trivial, making it possible to disregard $\varphi_{l}$ in Eq. (100). The
resulting system of evolution equations for the real amplitudes, including the
usual linear gain, $\gamma_{0}>0$, and cubic loss (cf. Eq. (1)), is

$\frac{dA_{l}}{dt}=\left(\gamma_{0}-\sum_{m=1}^{N}T_{l-m}A_{m}^{2}\right)A_{l}\equiv-\frac{\partial
L}{\partial A_{l}},$ (101)

where $T_{l-m}>0$ are coefficients of the cubic lossy nonlinearity, subject to
normalization $T_{0}=1$, and the Lyapunov function is

$L=-\frac{\gamma_{0}}{2}\sum_{l=1}^{N}A_{l}^{2}+\frac{1}{4}\sum_{l,m=1}^{N}T_{l-m}A_{l}^{2}A_{m}^{2},$
(102)

cf. Eqs. (2) and (3). Detailed analysis of the results, produced with the help
of Eq. (101), was presented in Ref. [10]), see also some preliminary results
in Refs. [97] and [98].

Spatially quasiperiodic patterns of the octagonal ($N=4$) and decagonal
($N=5$) types are displayed, respectively, in Figs. 6(a) and 7(a). It is seen
that they are built as compositions of rhombuses of different shapes, and the
presence of the overall octagonal or pentagonal structure is evident.

Solutions of Eq. (101) for $N=4$ depend on two independent nonlinearity
coefficients, $T_{1}=T_{3}$ and $T_{2}$. In this case, there are four distinct
stationary solutions which have their stability areas: rolls, with

$A_{1}=\sqrt{\gamma_{0}},A_{2,3,4}=0;$ (103)

a square lattice, with

$A_{1,3}=\sqrt{\gamma_{0}/\left(1+T_{2}\right)},A_{2,4}=0;$ (104)

an anisotropic rectangular lattice, with the aspect ratio
$\tan(\pi/8)=\allowbreak\sqrt{2}-1\approx\allowbreak 0.414$ and amplitudes

$A_{1,2}=\sqrt{\gamma_{0}/\left(1+T_{1}\right)},A_{3,4}=0;$ (105)

and the octagonal quasicrystal, with

$A_{1,2,3,4}=\sqrt{\gamma_{0}/\left(1+2T_{1}+T_{2}\right)}.$ (106)

In addition to that, Eq. (101) also has a stationary semi-periodic solution,
which is quasiperiodic in direction $\mathbf{n}_{2}$ and periodic along
$\mathbf{n}_{4}$, with $A_{1}=A_{2}\neq 0,A_{3}\neq 0,$ and $A_{4}=0$.
However, the latter solution is completely unstable.

The full stability chart for stationary solutions (103)-(106) can be readily
found in an analytical form. It is displayed in Fig. 6(b).

Figure 6: (a) The shape of the octagonal quasiperiodic pattern, reprinted from
Ref. [99]. (b) The stability chart for patterns composed of four amplitudes,
$A_{1,2,3,4}$, in the plane of nonlinearity coefficients, $T_{1}$ and $T_{2}$,
of Eq. (101), reprinted from Ref. [10]. Stability areas of the rolls (103),
squares (104), rectangles (105), and octagonal quasicrystal (106) are denoted
by encircled numbers 1, 2, 3, and 4, respectively.

For $N=5$, solutions of Eq. (101) also depend on two independent nonlinearity
coefficients, $T_{1}=T_{4}$ and $T_{2}=T_{3}$. These equations produce six
different species of stable stationary patterns: rolls, with

$A_{1}=\sqrt{\gamma_{0}},A_{2,3,4,5}=0;$ (107)

two different species of rectangular lattices:

$\displaystyle A_{1,2}$ $\displaystyle=$
$\displaystyle\sqrt{\gamma_{0}/\left(1+T_{1}\right)},A_{3,4,5}=0,$ (108)
$\displaystyle A_{1,3}$ $\displaystyle=$
$\displaystyle\sqrt{\gamma_{0}/\left(1+T_{2}\right)},A_{2,4,5}=0;$ (109)

the decagonal quasicrystal:

$A_{1,2,3,4,5}=\sqrt{\gamma_{0}/\left(1+2T_{1}+2T_{2}\right)};$ (110)

and two species of semi-periodic patterns, that are quasiperiodic in one
direction and periodic in the other:

$\displaystyle A_{1,3}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\gamma_{0}\left(1-T_{1}\right)}{1+T_{2}-2T_{1}^{2}}},A_{2}=\sqrt{\frac{\gamma_{0}\left(1+T_{2}-2T_{1}\right)}{1+T_{2}-2T_{1}^{2}}},A_{4,5}=0,$
(111) $\displaystyle A_{1,5}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\gamma_{0}\left(1-T_{2}\right)}{1+T_{1}-2T_{2}^{2}}},A_{3}=\sqrt{\frac{\gamma_{0}\left(1+T_{1}-2T_{2}\right)}{1+T_{1}-2T_{2}^{2}}},A_{2,4}=0.$
(112)

In addition to that, there is another semi-periodic solution, with
$A_{1}=A_{2}\neq 0$, $A_{3}=A_{4}\neq 0$, and $A_{5}=0$, but it is completely
unstable.

The full stability chart for this set of solutions was found too in an
analytical form, as shown in Fig. 7(b). In this figure, the constants are

$\omega_{1}=\frac{\sqrt{5}-1}{2}\approx 0.618,\omega_{2}=\omega_{1}+1.$ (113)

Note that, unlike the situation for the octagonal setting ($N=4$), displayed
in Fig. 6(b), the stability areas for the decagonal ($N=5$) quasicrystal (110)
and periodic patterns (108), (109), are not adjacent to each other in Fig.
7(b), being separated by regions of stable semi-periodic states (111) and
(112) (recall that all semi-periodic states are unstable in the case of
$N=4$).

Figure 7: (a) The shape of the decagonal quasiperiodic pattern, reprinted from
Ref. [99]. (b) The stability chart for patterns composed of five amplitudes,
$A_{1,2,3,4,5}$, in the plane of nonlinearity coefficients, $T_{1}$ and
$T_{2}$, of the respective system of equations (101), reprinted from Ref.
[10]. Stability areas of the rolls (107), rectangles (108) and (109),
decagonal quasicrystal (110), and semi-periodic states (111) and (112), are
denoted by encircled numbers 1, 2, 3, 4, 5, and 6, respectively. Constants
$\omega_{1}$ and $\omega_{2}$, marked in panel (b), are defined by Eq. (113).

It is relevant to mention that, if higher-order nonlinear terms are added to
the system of equations (101), the sharp boundaries between stability areas of
different patterns in Figs. 6(b) and 7(b) may be modified. In particular,
there may appear narrow strips of bistability (which is impossible in the
framework of Eq. (101)), as well as strips populated by more complex patterns,
instead of the sharp lines [10].

###  III.2 Dodecagonal quasicrystals ($N=6$)

In the case of the twelve-mode patterns, corresponding to $N=6$ in Eq. (101),
quadratic nonlinearity, with coefficient $\nu>0$, should be included too, as
the corresponding set of six wave vectors contains two resonant triads, that
may be naturally coupled by the quadratic terms (cf. Eq. (47)):

$\mathbf{n}_{1}+\mathbf{n}_{5}+\mathbf{n}_{9}=\mathbf{n}_{2}+\mathbf{n}_{6}+\mathbf{n}_{10}=0$
(114)

(recall that only six wave vectors are actually different, according to Eq.
(99)). In this case, the dynamics of phases of the complex amplitudes (6)
cannot be disregarded, giving rise to phason modes [100, 101, 102].
Accordingly, Eq. (101) is replaced by a coupled system of evolution equations
for the real amplitudes and phases [10]:

$\displaystyle\frac{dA_{l}}{dt}=\left(\gamma_{0}-\sum_{m=1}^{N}T_{l-m}A_{m}^{2}\right)A_{l}+\nu
A_{n+4}A_{n+8}\cos\Phi_{n},$ (115) $\displaystyle
A_{l}\frac{d\varphi_{l}}{dt}=-\nu A_{l+4}A_{l+8}\sin\Phi_{l},$ (116)
$\displaystyle\Phi_{l}\equiv\varphi_{l}+\varphi_{l+4}+\varphi_{l+8},$ (117)

with $\varphi_{l}\equiv\varphi_{l-12}$ for $l>12$.

These equations give rise to the following stationary solutions for the
dodecagonal quasicrystals, with equal values of real amplitudes $A_{l}$, and
the same values of $\Phi_{l}$ for both resonantly coupled triads (114):

$\displaystyle\cos\Phi_{l}=\pm 1,$ (118) $\displaystyle
A_{l}=\pm\left(2Q_{0}\right)^{-1}\left(\nu\pm\sqrt{\nu^{2}+4\gamma_{0}Q_{0}}\right),$
(119) $\displaystyle Q_{0}\equiv 1+2T_{1}+2T_{2}+T_{3}.$ (120)

The analysis of the stability of these solutions in the framework of Eqs.
(115)-(117) demonstrates that they may be stable only under conditions
$Q_{0}>0$ (see Eq. (120) and

$Q_{3}\equiv 1+T_{3}-T_{1}-T_{2}>0.$ (121)

If these conditions hold, the amplitude of stable quasicrystals exceeds a
minimum value,

$A_{l}\geq A_{\min}\equiv(1/2)\max\left\\{\nu/Q_{0},\nu/Q_{3}\right\\}.$ (122)

Further, there is no stability constraint from above for the amplitude,
provided that the following combinations of the nonlinearity coefficients are
positive:

$Q_{1,2}\equiv 1-T_{2}\pm\left(T_{1}-T_{3}\right)\geq 0.$ (123)

Otherwise, the stability imposes the following limit on the amplitude:

$A\leq A_{\max}\equiv\min\left\\{-\nu/Q_{1},-\nu/Q_{2}\right\\}$ (124)

(if only one combination $Q_{1}$ or $Q_{2}$ is negative, only this one
determines the upper limit for the stability, as per Eq. (124)).

The existence and stability results for the amplitude of the dodecagonal
quasicrystals is summarized in Fig. 8(b). Note that, in the presence of the
resonant interaction mediated by the quadratic term in Eq. (115), the solution
appears as a subcritical [103] one, with a finite value of the amplitude, at
$\gamma_{0}<0$, i.e., when this coefficient represents linear loss, rather
than gain.

Figure 8: (a) The shape of the dodecagonal quasiperiodic pattern, reprinted
from Ref. [99]. (b) The pattern’s amplitude, given by Eq. (119), vs. the
strength of the linear gain ($\gamma_{0}>0$) or loss ($\gamma_{0}<0$),
reprinted from Ref. [10]. Stable and unstable solutions are represented,
respectively, by bold and thin lines. Branches 1, 2 and 3 pertain, severally,
to the solutions with $\cos\Phi_{l}=+1$ and $-1$ in Eq. (118). Parameters
$Q_{0}$, $A_{\min}$, and $A_{\max}$ are defined as per Eqs. (120), (122), and
(124), respectively.

###  III.3 A quasicrystalline layer between orthogonally oriented square-
lattice patterns

While, as shown in Fig. 6(b), square-lattice and octagonal quasiperiodic
patterns cannot coexist as stable ones in the system with $N=4$, it was
demonstrated in Ref. [83] that a sufficiently broad stripe filled by an
effectively stable quasiperiodic pattern may be realized as a transient layer
between stable square-lattice patterns mutually oriented under the angle of
$45^{\mathrm{o}}$, as schematically shown in Fig. 9(a). For this configuration
(which naturally combines the two main topics of the present article, viz.,
the DWs and QP patterns), one may naturally adopt equal amplitudes
corresponding to wave vectors $\mathbf{k}_{3}$ and $\mathbf{k}_{4}$,

$A_{3}(x)=A_{4}(x)\equiv A(x),$ (125)

while amplitudes $B_{1}$ and $B_{2}$ related to $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}$ are different, the effective diffusion coefficient for the
latter one being zero, as per Eq. (43). The corresponding system of stationary
real GL equations, naturally extending Eqs. (52), (53), (61), (62), and (101),
takes the following form:

$\displaystyle\frac{1}{2}\frac{d^{2}A}{dx^{2}}+A-A^{3}-\left(T_{1}B_{1}^{2}+T_{1}B_{2}^{2}+T_{2}A^{2}\right)A=0,$
(126)
$\displaystyle\frac{d^{2}B_{1}}{dx^{2}}+B_{1}-B_{1}^{3}-\left(2T_{1}A^{2}+T_{2}B_{2}^{2}\right)B_{1}=0,$
(127) $\displaystyle
B_{2}-B_{2}^{3}-\left(2T_{1}A^{2}+T_{2}B_{1}^{2}\right)B_{2}=0,$ (128)

where, like in Eq. (101), $T_{1}$ and $T_{2}$ are coefficients of the cross-
interaction between the PW modes with angles, respectively, $45^{\mathrm{o}}$
and $90^{\mathrm{o}}$ between their wave vectors. According to Fig. 6(b), the
stability conditions for the spatially uniform square-lattice and octagonal
quasicrystalline patterns are, respectively,

$\displaystyle T_{2}$ $\displaystyle\leq$ $\displaystyle 1,T_{1}\geq
T_{2}+1/2,$ (129) $\displaystyle T_{2}$ $\displaystyle\leq$ $\displaystyle
1,T_{1}\leq T_{2}+1/2.$ (130)

Accordingly, to secure the stability of the background square lattices and a
possibility to have a broad transient layer between mutually rotated ones,
which is filled by the effectively stable octagonal pattern, it is relevant to
choose parameters belonging to the stability area (129), with values close to
the stability boundary, $T_{1}=T_{2}+1/2$. An appropriate choice is

$T_{1,2}\equiv 1-\mu_{1,2},~{}0<\mu_{1,2}\ll 1,$ (131) $\text{with }m\equiv
2\mu_{1}/\mu_{2}<1.$ (132)

Similar to what is considered above in Eqs. (63) and (64), Eq. (128) obviously
splits in two options: $B_{2}=0$, or

$B_{2}^{2}+2T_{1}A^{2}+T_{2}B_{1}^{2}=1.$ (133)

In either case, Eqs. (126) and (127) simplify accordingly. The solutions
corresponding to $B_{2}=0$ or to Eq. (133) must be “dovetailed” at a stitch
point $x=x_{0}$, cf. Eq. (65). An example of the so obtained solutions for
amplitudes $A(x)$ and $B_{1,2}(x)$ is displayed in Fig. 9(b).

Figure 9: (a) The scheme for building a broad stripe of the octagonal
quasicrystalline state as a transient layer between semi-infinite domains
filled by square-lattice patterns, mutually rotated by $45^{\mathrm{o}}$. (b)
An example of the corresponding solution for amplitudes $A(x)$ and
$B_{1,2}(x)$. Parts of the solution corresponding to Eq. (133) or to $B_{2}=0$
are connected at the stitch point $x=x_{0}$. Reprinted from Ref. [83].

###  III.4 Three-dimensional quasicrystals

A setting which makes it possible to predict a stable quasiperiodic pattern,
based on a set of four PW modes, in the 3D space was put forward in Ref. [82].
It originates from the model of a lasing cavity, based on the standard system
of coupled Maxwell-Bloch equations. The evolutional variable in this system is
time, while the spatial structure is strongly anisotropic, as the field
(Maxwell’s) equation in the system contains only the first derivative,
$\partial/\partial z$, with respect to the longitudinal coordinate, $z$, and
the usual paraxial-diffraction operator,
$i\left(\partial_{x}^{2}+\partial_{y}^{2}\right),$acting on the transverse
coordinates, $\left(x,y\right)$. As a result, at the lasing threshold
components of 3D wave vectors carrying the PW modes,

$\mathbf{K}=(\mathbf{k},k_{z}),~{}\mathbf{k}\equiv\left(k_{x},k_{y}\right),$
(134)

satisfy the following dispersion relation, which couples them to the wave’s
frequency $\Omega$:

$\Omega=k^{2}+k_{z}.$ (135)

Eventually, above the lasing threshold the cubic nonlinearity of the Maxwell-
Bloch system may produce a _resonant quartet_ of 3D wave vectors, coupled by
condition

$\mathbf{K}_{1}+\mathbf{K}_{2}=\mathbf{K}_{3}+\mathbf{K}_{4}.$ (136)

For comparison, in the 2D space the same relation (136), taken close to the
threshold, i.e., for nearly equal length of the wave vectors, would imply that
the four vectors form a rhombus, and the cubic interaction between the
corresponding amplitudes, $u_{1,2,3,4}$, would be represented by usual
nonresonant nonlinear terms, essentially the same as in Eq. (101), with
$A_{l}$ replaced by $u_{l}$ and $A_{m}^{2}A_{l}$ replaced by the XPM terms,
$\left|u_{m}\right|^{2}u_{l}$. However, in the 3D setting the resonance
condition (136), combined with the dispersion relation (135), leads to a
nontrivial possibility to add four-wave-mixing (FWM) cubic terms to the XPM
ones, see below.

Substituting expression (134) for the 3D wave vector in Eqs. (136) and (135)
leads to the following elementary exercise in planar geometry: find two pairs
of 2D vectors, $\left(\mathbf{k}_{1},\mathbf{k}_{2}\right)$ and
$\left(\mathbf{k}_{3},\mathbf{k}_{4}\right)$, satisfying conditions

$\mathbf{k}_{1}+\mathbf{k}_{2}=\mathbf{k}_{3}+\mathbf{k}_{4},~{}k_{1}^{2}+k_{2}^{2}=k_{3}^{2}+k_{4}^{2}.$
(137)

An obvious solution of this exercise is plotted in Fig. 10(a).

Figure 10: (a) A set of four two-dimensional vectors $\mathbf{k}_{1,2,3,4}$
which solves equations (137). Here, A and B are two arbitrary points belonging
to the circumference of arbitrary radius $R$, and $a<R$ is an arbitrary value
of coordinate $x$. (b) An example of the three-dimensional quasiperiodic
pattern, projected onto the $\left(x,y\right)$ plane. Shown are contour plots
of the corresponding distribution of the order parameter,
$\mathrm{Re}\left[\sum_{l=1}^{4}u_{l}\cos\left(\mathbf{k}_{l}\cdot\mathbf{R}\right)\right]$,
where $u_{l}$ are the complex amplitudes given by Eqs. (141), (142), and
(143). In this case, the phases are $\varphi_{1}=\pi,\varphi_{2}=-\pi/2,$
$\varphi_{3}=\pi/7$, while $\varphi_{4}$ is determined by Eq. (142). The angle
between vectors $\mathbf{k}_{1}-\mathbf{k}_{2}$ and
$\mathbf{k}_{3}-\mathbf{k}_{4}$ is $\pi/5$. The figure is reprinted from Ref.
[82].

Once a resonantly coupled quartet of four wave vectors is chosen, the
respective system of evolution equations for the corresponding complex
amplitudes is [82]

$\displaystyle\frac{du_{1}}{dt}$ $\displaystyle=$
$\displaystyle\gamma_{0}u_{1}-\left(\left|u_{1}\right|^{2}+2\sum_{l\neq
1}\left|u_{l}\right|^{2}\right)u_{1}-2u_{2}^{\ast}u_{3}u_{4},$
$\displaystyle\frac{du_{2}}{dt}$ $\displaystyle=$
$\displaystyle\gamma_{0}u_{2}-\left(\left|u_{2}\right|^{2}+2\sum_{l\neq
2}\left|u_{l}\right|^{2}\right)u_{2}-2u_{1}^{\ast}u_{3}u_{4},$
$\displaystyle\frac{du_{3}}{dt}$ $\displaystyle=$
$\displaystyle\gamma_{0}u_{3}-\left(\left|u_{3}\right|^{2}+2\sum_{l\neq
3}\left|u_{l}\right|^{2}\right)u_{3}-2u_{1}u_{2}u_{4}^{\ast},$
$\displaystyle\frac{du_{4}}{dt}$ $\displaystyle=$
$\displaystyle\gamma_{0}u_{4}-\left(\left|u_{4}\right|^{2}+2\sum_{l\neq
4}\left|u_{l}\right|^{2}\right)u_{4}-2u_{1}u_{2}u_{3}^{\ast}.$

In these equations, $\gamma_{0}>0$ is the linear gain, as above, and the last
terms represent the above-mentioned FWM effect. Particular values of
coefficients in front of nonlinear terms are standard ones which correspond to
the XPM and FWM interactions in nonlinear optics [19], unlike general values
of coefficients $T_{l-m}$ in Eq. (101). Similar to Eq. (101), the system of
equations (LABEL:FWM) admits the presentation in the form of $du_{l}/dt=-$
$\partial L/\partial u_{l}^{\ast}$, with the Lyapunov function

$L=-\gamma_{0}\sum_{l}\left|u_{l}\right|^{2}+\frac{1}{2}\sum_{l}\left|u_{l}\right|^{4}+2\sum_{l>m}\left|u_{l}\right|^{2}\left|u_{m}\right|^{2}+4\mathrm{Re}\left(u_{1}u_{2}u_{3}^{\ast}u_{4}^{\ast}\right).$
(139)

Further analysis performed in Ref. [82] had produced two stable stationary
solutions of Eqs. (LABEL:FWM). First, this is a simple single-mode state
(rolls), with

$\left|u_{1}\right|^{2}=\gamma_{0},~{}u_{2,3,4}=0.$ (140)

Next, dodecagonal quasicrystals with equal absolute values of all the four
amplitudes are looked for as

$u_{l}=A\exp\left(i\varphi_{l}\right),$ (141)

where the phased are locked so that

$\varphi_{1}+\varphi_{2}-\varphi_{3}-\varphi_{4}=\pi,$ (142)

and the squared absolute value of the amplitudes is

$\left|u_{l}\right|^{2}=\gamma_{0}/5,$ (143)

cf. the rolls solution (140). Note that values of the Lyapunov function (139)
for the rolls and 3D quasicrystal are

$L_{\mathrm{rolls}}=-\gamma_{0}^{2}/2,~{}L_{\mathrm{quasicryst}}=-2\gamma_{0}^{2}/5,$
(144)

hence the rolls represent the ground state of the system, while the
quasicrystal is a metastable state, as its value of $L$ is slightly higher.

An example of the shape of the 3D quasiperiodic solution is displayed, in the
projection onto plane $\left(x,y\right)$, in Fig. 10(b). Additional examples
can be found in Ref. [82].

Besides these solutions, Eqs. (LABEL:FWM) give rise to another quasiperiodic
state, with $\varphi_{1}+\varphi_{2}-\varphi_{3}-\varphi_{4}=0$ and
$\left|u_{l}\right|^{2}=\gamma_{0}/9$ (cf. Eqs. (142) and (143)), but it is
unstable. Also exist but are unstable two-mode solutions, e.g., ones with
$\left|u_{1,2}\right|^{2}=\gamma_{0}/3,u_{3,4}=0$ [82].

##  IV Conclusion

The aim of this article is to present a concise overview of two important
topics in the theory of pattern formation in nonlinear dissipative media,
viz., DWs (domain walls) and QP (quasiperiodic) patterns. The topics are
selected as those important contributions to which were made in works of Prof.
Mikhail Tribelsky. Most results collected in this article may be considered as
rather “old” ones, as they had been published ca. 30-35 [9, 10, 12, 29, 39,
48, 51, 58, 65, 67, 68, 94, 98] or 20 [71, 82, 83, 84, 66] years ago.
Nevertheless, these results remain relevant in the context of ongoing
theoretical and experimental studies in the ever expanding pattern-formation
research area. This conclusion is upheld by the fact that the present article
includes a few novel exact analytical results, obtained as a relevant addition
to the old theoretical findings concerning the DWs in systems of coupled real
GL (Ginzburg-Landau) equations [11]. The new results, represented by Eqs.
(54), (63)-(68), (72)-(75), and (87)-(90) produce exact solutions for
symmetric DWs in the system of real GL equations including linear mixing
between the components, the solution for strongly asymmetric DWs in the case
when the diffusion term is present only in one GL equation, the three-
component composite state including the DW in two components and a bright
soliton in the third one, and the particular exact solution for DWs between
waves governed by the real GL equations including group-velocity terms with
opposite signs.

The significance of the results presented in this brief review is enhanced by
the fact that essentially the same coupled equations describe patterns of the
DW and QP types not only in thermal convection, but also in nonlinear optics,
BEC, and other physical systems. In particular, the pattern formation in BEC
of cesium atoms under the action of a temporally-periodic modulation of the
nonlinearity (imposed by means of the Feshbach resonance [104]), similar to
the Faraday instability, was recently experimentally demonstrated and
theoretically modeled in the framework of amplitude equations similar to Eqs.
(101) in Ref. [105]. Another novel realization of the pattern formation was
proposed for a driven dissipative Bose-Hubbard lattice, that can be
implemented in superconducting circuit arrays [106].

It is expected that theoretical and experimental studies along the directions
outlined in this article have a potential for further development, which will
make it possible to add new findings to the above-mentioned well-established
results.

## Acknowledgments

First of all, I would like to thank Mikhail Tribelsky for the collaboration,
established long ago, which had produced essential results summarized in this
article. I also thank other colleagues in collaboration with whom these
results were obtained: Alexander Nepomnyashchy, Jerry Moloney, Alan Newell,
Natalia Komarova, Martin Van Hecke, and Horacio Rotstein. I appreciate a
discussion of the topic of domain walls with Dmitry Pelinovsky, and the help
of Zhaopin Chen in producing Figs. 4 and 5.

As one of editors of the Special Issue of journal Physics (published by MDPI),
dedicated to the celebration of the 70th birthday of Mikhail Tribelsky, for
which this article was written, I thank two other editors of the Special
Issue, Andrey Miroshnichenko and Fernando Moreno, for very efficient
collaboration.

This work was supported, in part, by Israel Science Foundation through grant
No. 1286/17.

## References

  * [1] Anisimov, S. I.; Tribelsky, M. I.; Epelbaum, Y. G. Instability of a plane evaporation boundary in the interaction between laser radiation and matter. Zh. Eksp. Teor. Fiz. 1980, 78, 1597-1605.
  * [2] Bunkin F. V.; Tribelsky, M. I. Non-resonant interaction of high-power optical radiation with a liquid. Uspekhi Fiz. Nauk. 1980, 130, 193-239.
  * [3] Luk’yanchuk, B. S.; Tribelsky, M. I. Anomalous light scattering by small particles. Phys. Rev. Lett. 2006, 97, 263902.
  * [4] Tribelsky, M. I.; Flach, S.; Miroshnichenko, A. E.; Gorbach, A. V.; Kivshar, Y. S. Light scattering by a finite obstacle and Fano resonances, Phys. Rev. Lett. 2008, 100, 043903.
  * [5] Tribelsky, M. I.; Geffrin, J. M.; Litman, A.; Eyraud, C.; Moreno, F. Small dielectric spheres with high refractive index as new multifunctional elements for optical devices. Sci. Rep. 2015, 5, 12288.
  * [6] Miroshnichenko, A. E.; Tribelsky, M. I. Giant in-particle field concentration and Fano resonances at light scattering by high-refractive-index particles. Phys. Rev. A. 2016, 83, 053837.
  * [7] Miroshnichenko, A. E.; Tribelsky, M. I. Ultimate absorption in light scattering by a finite obstacle. Phys. Rev. Lett. 2018, 120, 263902.
  * [8] Name known as Mikhail/Michael/Michel/Miguel/Michele/Michal/Mikael/Mikkel/Mitxel/… is considered as the oldest masculine name used in modern languages. The meaning of its original form in Hebrew is “who (is) like El (God)?”, which implies a response “no one can be compared to God”.
  * [9] Malomed, B. A.; Nepomnyashchy, A. A.; Tribelsky, M. I. Domain boundaries in convection patterns. Phys. Rev. A 1990, 42, 7244-7263.
  * [10] Malomed, B. A.; Nepomnyashchy, A. A.; Tribelsky, M. I. Two-dimensional quasiperiodic patterns in nonequilibrium systems. Zh. Eksp. Teor. Fiz. 1989, 96, 688-700.
  * [11] Malomed, B. A. New findings for the old problem: Exact solutions for domain walls in coupled real Ginzburg-Landau equations, Phys. Lett. A, in press.
  * [12] Malomed, B. A.; Tribelsky, M. I. Bifurcations in distributed kinetic systems with aperiodic instability. Physica D 1984, 14, 67-87.
  * [13] Ginzburg, V. L. and Landau, L. D. On the theory of superconductivity. Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki (USSR) 1950, 20, 1064–1082 (in Russian) [English translation: in Men of Physics, vol. 1. 1965. Oxford: Pergamon Press, pp. 138–167].
  * [14] Cross, M. C. Ingredients of a theory of convective textures close to onset. Phys. Rev. A 1982, 25, 1065-1076.
  * [15] Eckhaus, W. Studies in Non-Linear Stability Theory, Springer, New York, 1965.
  * [16] Wiktor Eckhaus (1930-2000) was born in Poland, where he had survived Holocaust. After WWII, he moved to the Netherlands, where he had eventually become a professor at the Utrecht University.
  * [17] Aranson, I. S.; Kramer, L. The world of the complex Ginzburg-Landau equation. Reviews of Modern Physics. 2002, 74, 99-143.
  * [18] Malomed, B. A. Complex Ginzburg-Landau equation. In: Encyclopedia of Nonlinear Science, pp. 157-160. A. Scott, editor. New York: Routledge, 2005.
  * [19] Kivshar, Y. S.; Malomed, B. A. Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 1989, 61, 763-915.
  * [20] Arecchi, F. T.; Boccaletti, S.; Ramazza, P. Pattern formation and competition in nonlinear optics. 1999, Physics Reports, 318, 1–83.
  * [21] Rosanov, N. N. Transverse patterns in wide-aperture nonlinear optical systems. Progr. Opt. 1996, 35, 1-60.
  * [22] Rosanov, N. N. Spatial Hysteresis and Optical Patterns. Springer-Verlag, Berlin, 2002.
  * [23] Lega, J. Traveling hole solutions of the complex Ginzburg-Landau equation: a review. Physica D 2001, 152, 269–287.
  * [24] Cross, M. C.; Hohenberg, P. C. Pattern-formation outside of equilibrium. Reviews of Modern Physics 1993, 65: 851-1112.
  * [25] Ipsen, M.; Kramer, L.; Sorensen, P. G. Amplitude equations for description of chemical reaction-diffusion systems.  Phys. Rep. 2000, 337, 193–235.
  * [26] Hoyle, R. Pattern Formation: An Introduction to Methods. Cambridge University Press, Cambridge, 2006.
  * [27] Malomed, B. A. Strong periodic amplification of solitons in a lossy optical fiber: Analytical results. J. Opt. Soc. Am. B 1994, 11, 1261-1266.
  * [28] Berntson, A.; Malomed, B. A. Dispersion-management with filtering. Opt. Lett. 1999, 24, 507-509.
  * [29] Malomed, B. A. Evolution of nonsoliton and “quasiclassical” wavetrains in nonlinear Schrödinger and Korteweg - de Vries equations with dissipative perturbations. Physica D 1987, 29, 155-172.
  * [30] Sakaguchi, H. Motion of pulses and vortices in the cubic-quintic complex Ginzburg-Landau equation without viscosity. Physica D 2005, 210, 138-148.
  * [31] Akhmediev N.; Ankiewicz A., Editors. Dissipative Solitons: From Optics to Biology and Medicine, Springer-Verlag, Springer, 2008.
  * [32] Wise, F. W.; Chong, A.; Renninger, W. H. High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion. Laser Phot. Rev. 2008, 2, 58-73.
  * [33] T. Ackemann; Firth, W. J.; Oppo, G. L. Fundamentals and applications of spatial dissipative solitons in photonic devices. Adv. At. Mol. Opt. Phys. 2009, 57, 323-421.
  * [34] Leblond, H.; Mihalache, D. Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys. Rep. 2013, 523, 61-126.
  * [35] Song, Y. F.; Shi, X. J.; Wu, C. F.; Tang, D. Y.; Zhang, H. Recent progress of study on optical solitons in fiber lasers. Appl. Phys. Rev. 2019, 6, 0213139.
  * [36] Hocking L. M.; Stewartson, K. On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. R. Soc. London Ser. A 1972, 326, 289-313.
  * [37] Pereira N. R.; Stenflo, L. Nonlinear Schrödinger equation including growth and damping. Phys. Fluids 1977, 20, 1733-1734.
  * [38] Bekki, N.; Nozaki, K.. Formation of spatial patterns and holes in the generalized Ginzburg-Landau equation. Phys. Lett. A 1985. 110, 133-135.
  * [39] Malomed, B. A. Stability and grain boundaries in the dispersive Newell-Whitehead-Siegel equation. Physica Scripta 1997, 57, 115-117.
  * [40] Lugiato, A. A.; Lefever R. Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 1987, 58, 2209-2211.
  * [41] Oppo G.-L.; Brambilla, M.; Lugiato, L. Formation and evolution of roll patterns in optical parametric oscillators. Phys. Rev. A 1994, 49, 2028-2032.
  * [42] Chembo, Y. K.; Menyuk, C. R. Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators. Phys. Rev. A 2013, 87, 053852.
  * [43] Huang, S. W.; Yang, J. H.; Yang, S. H.; Yu, M. B.; Kwong, D. L.; Zelevinsky, T.; Jarrahi, M.; Wong, C. W. Globally stable microresonator Turing pattern formation for coherent high-power THz radiation on-chip. Phys. Rev. X 2017, 7, 041002.
  * [44] de Valcarcel, G. J.; Staliunas, K. Phase-bistable Kerr cavity solitons and patterns. Phys. Rev. A 2013, 87, 043802.
  * [45] Garbin, B.; Wang, Y. D.; Murdoch, S. G.; Oppo, G. L.; Coen, S.; Erkintalo, M. Experimental and numerical investigations of switching wave dynamics in a normally dispersive fibre ring resonator. Eur. Phys. J. D 2017, 71, 240.
  * [46] Mineev, V. P. The theory of the solution of two near-ideal Bose gases. Zh. Eksp. Teor. Fiz. 1974, 67, 263-272 (1974) [English translation: Sov. Phys. – JETP 1974, 40, 132-136].
  * [47] Petviashvili, V. I.; Sergeev, A. M. Spiral solitons in active media with an excitation threshold. Doklady Akademii Nauk SSSR 1984, 276, 1380–1384 (in Russian).
  * [48] Fauve, S.; Thual, O. Solitary waves generated by subcritical instabilities in dissipative systems. Phys. Rev. Lett. 1990, 64, 282-284.
  * [49] van Saarloos W.; Hohenberg P. C. Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation, Phys. Rev. Lett. 1990, 84, 749-752.
  * [50] Hakim, V.; Jakobsen, P.; Pomeau, Y. Fronts vs. solitary waves in nonequilibrium systems. Europhys. Lett. 1990, 11, 19-24.
  * [51] Malomed, B. A.; Nepomnyashchy, A. A. Kinks and solitons in the generalized Ginzburg-Landau equation. Phys. Rev. A 1990, 42, 6009-6014.
  * [52] Kuramoto, Y.; Tsuzuki, T. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progr. Theor. Phys. 1976, 55, 356-369.
  * [53] Sivashinsky, G. I. Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations. Acta Astronautica 1977, 4, 1177–1206.
  * [54] Kuramoto, Y. Diffusion-induced chaos in reaction systems. Progr. Theor. Phys. Supplement 1978, 64, 346–367.
  * [55] Cladis, P. E.; Fradin, C.; Finn, P. L.; Brand, H. R. A novel route to defect turbulence in nematics. Mol. Cryst. Liq. Cryst. Sci. Tech. A: Mol. Cryst. Liq. Cryst. 1998, 328, 513-521.
  * [56] Manneville, P.; Pomeau, Y. A grain-boundary in cellular structures near the onset of convection. Phil. Mag. A 1983, 48 607-621.
  * [57] Haragus, M.; Iooss, G. Bifurcation of symmetric domain walls for the Bénard-Rayleigh convection problem. Arch. Rational Mech. Anal. 2021, 239, 733-781.
  * [58] Steinberg, V.; Ahlers, G.; Cannell, D. S. Pattern formation and wave-number selection by Rayleigh-Bénard convection in a cylindrical container. Physica Scripta 1985, 32, 534-547.
  * [59] Rohrer, G. S. Grain boundary energy anisotropy: a review. J. Materials Science 2011, 46, 5881-5895.
  * [60] Lim, H.; Lee, M. G.; Wagoner, R. H. Simulation of polycrystal deformation with grain and grain boundary effects. Int. J. Plasticity 2011, 27, 1328-1354.
  * [61] Rudolph, P. Dislocation patterning and bunching in crystals and epitaxial layers - a review. Cryst. Res. Tech. 2017, 52, 1600171.
  * [62] Atxitia, U.; Hinzke, D.; Nowak, U. Fundamentals and applications of the Landau-Lifshitz-Bloch equation. J. Phys. D: Appl. Phys. 2017, 50, 033003.
  * [63] Galkina, E. G.; Ivanov, B. A. Dynamic solitons in antiferromagnets. Low Temp. Phys. 2018, 44, 618-633 Published: JUL 2018
  * [64] Yao, W.; Wu, B.; Liu, Y. Growth and grain boundaries in 2D materials. ACS NANO 2020, 14, 9320-9346.
  * [65] Malomed, B. A. Optical domain walls. Phys. Rev. E 1994, 50, 1565-1571.
  * [66] Trippenbach, M.; Góral, K.; Rzażewski, K.; Malomed. B.; Band, Y. B. Structure of binary Bose-Einstein condensates. J. Phys. B: At. Mol. Opt. Phys. 2000, 33, 4017-4031.
  * [67] Malomed, B. A. Nonsteady waves in distributed dynamical systems. Physica D 1983, 8, 353-359.
  * [68] Cross, M. C. Traveling and standing waves in binary-fluid convection in finite geometries. Phys. Rev. Lett. 1986, 57, 2935-2938.
  * [69] Cross, M. C. Structure of nonlinear traveling-wave states in finite geometries, Phys. Rev. A 1988, 38, 3593-3600 (1988).
  * [70] Coullet, P.; Frisch, T.; Plaza, F. Sources and sinks of wave patterns. Physica D 1993, 62, 75-79.
  * [71] Voss, H. U.; Kolodner, P.; Abel, M.; Kurths, J. Amplitude equations from spatiotemporal binary-fluid convection data. Phys. Rev. Lett. 1999, 83, 3422-3425.
  * [72] Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 1984, 53, 1951-1953.
  * [73] Wang, N.; Chen, H.; Kuo, K. Two-dimensional quasicrystal with eightfold rotational symmetry. Phys. Rev. Lett. 1987, 59, 1010-1013.
  * [74] Torquato, S. Hyperuniform states of matter. Phys, Rep. 2018, 745, 1-95.
  * [75] Steurer, W. Quasicrystals: What do we know? What do we want to know? What can we know? Acta Crystallographica A 2018, 74, 1-11.
  * [76] Skjaervo, S. H.; Marrows, C. H.; Stamps, R. L.; Leyderman, L. J. Advances in artificial spin ice. Nature Rev. Phys. 2020, 2, 13-28.
  * [77] von Freymann, G; Ledermann, A; Thiel, M.; Staude, I.; Essig, S.; Busch, K.; Wegener, M. Three-dimensional nanostructures for photonics. Adv. Functional Materials 2010, 20, 1038-1052.
  * [78] Bellingeri, M.; Chiasera, A.; Kriegel, I.; Scotognella, F. Optical properties of periodic, quasi-periodic, and disordered one-dimensional photonic structures. Optical Materials 2017, 72, 403-421.
  * [79] Vardeny, Z. V.; Nahata, A.; Agrawal, A. Optics of photonic quasicrystals. Nature Phot. 2013, 7, 177-187.
  * [80] Lu, L.; Joannopoulos, J. D.; Soljacic, M. Topological photonics. Nature Phot. 2014, 8, 821-829.
  * [81] Steurer, W.; Sutter-Widmer, D. Photonic and phononic quasicrystals. J. Phys. D: Appl. Phys. 2007, 40, R229-R247.
  * [82] Komarova, N.; Malomed, B. A.; Moloney, J. V.; Newell, A. C. Resonant quasiperiodic patterns in a three-dimensional lasing medium. Phys. Rev. A 1997, 56, 803-812.
  * [83] Rotstein, H.; Malomed, B. A. A quasicrystallic domain wall in nonlinear dissipative patterns. Physica Scripta 2000, 62, 164-168.
  * [84] van Hecke, M.; Malomed, B. A. A domain wall between single-mode and bimodal states and its transition to dynamical behavior in inhomogeneous systems. Physica D 1997, 101, 131-156.
  * [85] Kivshar, Y. S.; G. P. Agrawal. Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
  * [86] Skorobogatiy, M.; Yang, J. Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, 2009).
  * [87] Pitaevskii, L. P.; Stringari, S. Bose-Einstein Condensation (Oxford University Press, Oxford, 2003).
  * [88] Ballagh, R. J.; Burnett, K.; Scott, T. F. Theory of an output coupler for Bose-Einstein condensed atoms. Phys. Rev. Lett. 1997, 78, 1608-1611.
  * [89] Asghari, M.; White, I. H.; Penty, R. V. Wavelength conversion using semiconductor optical amplifiers. J. Lightwave Tech. 1997, 15, R3310-R3313.
  * [90] Kim, J.; Laemmlin, M.; Meuer, C.; Bimberg, D.; Eisenstein, G. Theoretical and experimental study of high-speed small-signal cross-gain modulation of quantum-dot semiconductor optical amplifiers. IEEE J. Quant. Elect. 2009, 45, 240-248.
  * [91] Merhasin, M. I.; Malomed, B. A.; Driben, R. Transition to miscibility in a binary Bose-Einstein condensate induced by linear coupling. J. Phys. B: At. Mol. Opt. Phys. 2005, 38, 877-892.
  * [92] Alama, S.; Bronsard, L.; Contreras, A.; Pelinovsky, D. E. Domains walls in the coupled Gross-Pitaevskii equations. Arch. Rat. Mech. Appl. 2015, 215, 579-615.
  * [93] Busse, F. H. The stability of finite amplitude cellular convection and its relation to an extremum principle. J. Fluid Mech. 1967, 30, 625-649.
  * [94] Pomeau Y. Front motion, metastability and subcritical bifurcations in hydrodynamics. Physica D 1986, 23, 3-11.
  * [95] Marzok, C.; Deh, B.; Courteille, P. W.; Zimmermann, C. Ultracold thermalization of $7$Li and 87Rb. Phys. Rev. A 200, 76, 052704.
  * [96] Malomed, B. A. Domain wall between traveling waves. Phys. Rev. E 1994, 50, R3310-R3313.
  * [97] Malomed, B. A.; Tribelsky, M. I. On the stability of stationary weakly overcritical patterns in convection and allied problems. Zh. Eksp. Teor. Fiz. (Sov. Phys. JETP) 1987, 92, 539-548.
  * [98] Malomed, B. A.; Nepomnyashchy, A. A.; Tribelsky, M. I. Two-dimensional dissipative structures with a quasicrystallic symmetry. Pis’ma Zh. Tekh. Fiz. (Sov. Phys. Tech. Phys. Lett.) 1987, 13, 1165-1167.
  * [99] Steurer, W. Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals. Zeitschrift für Kristallographie. 2004, 219, 391-446.
  * [100] Socolar, J. E. S.; Lubensky, T. C.; Steinhardt, P. J. Phonons, phasons, and dislocations in quasi-crystals. Phys. Rev. B 1986, 34, 3345-3360.
  * [101] Yamamoto, A. Crystallography of quasiperiodic crystals. Acta Crystallographica A 1996, 52, 509-560.
  * [102] Freedman, B.; Lifshitz, R.; Fleischer, J. W.; Segev, M. Phason dynamics in nonlinear photonic quasicrystals. Nature Materials 2007, 6, 776-781.
  * [103] Iooss, G.; Joseph, D. D. Elementary Stability Bifurcation Theory (Springer, New York, 1980).
  * [104] Chin, C.; Grimm, R.; Julienne, P.; Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 2010, 82, 1225-1286.
  * [105] Zhang, Z.; Yao, K.-X.; Feng, L., Hu, J.; Chin, C. Pattern formation in a driven Bose–Einstein condensate. Nature Phys. 2020, 16, 652-656.
  * [106] Wang, Z.; Navarrete-Benlloch, C.; Cai, Zi. Pattern formation and exotic order in driven-dissipative Bose-Hubbard systems. Phys. Rev. Lett. 2020, 125, 115301.