Preface

The nonlinear Schrödinger equation is a well-known equation that arises for a wide range of scientific purposes, including optical fiber communication systems, quantum mechanics, thermodynamics, ocean acoustic performance, biomedical dynamics, and quantum physics.

In this book, we present exact and numerical solutions to the nonlinear Schrödinger equation and its applications from various perspectives.

This book is a collection of selectively chosen chapters written by some of the world's leading researchers in quantum mechanics and nonlinear optics, particularly concerning the nonlinear Schrödinger equation.

Chapter 1 investigates the existence and stability properties of fundamental lattice solitons in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity.

In Chapter 2, linear stability analysis is used to study the modulation instability gain for a generalized nonlinear Schrödinger equation with rational nonlinear terms.

Chapter 3 presents different types of soliton solutions, such as bright, dark, singular, and W-shaped solitons, for the extended non-trivial version of the nonlinear Schrödinger equation. The Adomian decomposition method is used to compare the soliton solutions obtained using the indeterminate coefficient method.

In Chapter 4, the Projective Riccati equation technique is used to find various types of exact resonant optical soliton solutions, such as bright, dark, singular, king, dark-singular, and combined singular solitons, for the (3 + 1) dimensional resonant nonlinear Schrödinger equation with Kerr and parabolic nonlinearities.

Chapter 5 investigates the existence and stability properties of traveling wave solutions for the perturbed nonlinear Schrödinger equation with power-law nonlinearity and higher-order dispersions in a nano-optical fiber.

Chapter 6 discusses the non-Manakovian transmission phenomena for nonlinear depolarization of light governed by coupled nonlinear Schrödinger equation in optical fiber.

Chapter 7 presents a set of generalized Schrödinger's equations using the Hamilton– Jacobi equation and lifting principle. The classical Schrödinger's equation is demonstrated to be the simplest of this set.

Finally, Chapter 8 investigates the paradox of Schrödinger's cat and double-sit postulation. The author gives a different perspective on questions about the Schrödinger equation and the quantum conspiracy that physicists have debated for many years.

This book is a useful resource for scientists, researchers, and postgraduate students from various backgrounds.

> **Dr. Nalan Antar and Dr. İlkay Bakırtaş** Department of Mathematics Engineering, Istanbul Technical University, İstanbul, Türkiye

### **Chapter 1**

## Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing and Self-Defocusing Quintic Nonlinearity

*Mahmut Bağcı,Theodoros P. Horikis, İlkay Bakırtaş and Nalan Antar*

### **Abstract**

The fundamental lattice solitons are explored in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity. The band-gap boundaries, soliton profiles, and stability domains of fundamental solitons are investigated comprehensively by the linear stability spectra and nonlinear evolution of the solitons. It is demonstrated that fundamental lattice solitons can stay stable for a wide range of parameters with the weak self-focusing and self-defocusing quintic nonlinearity, while strong self-focusing and self-defocusing quintic nonlinearities are shortened the propagation distance of evolved solitons. Furthermore, it is observed that when the instability emerges from strong quintic nonlinearity, increasing anisotropy of the medium and modification of lattice depth can be considered as a collapse arrest mechanism.

**Keywords:** lattice solitons, nonlinear response, nonlocal nonlinear medium, quintic nonlinearity, focusing and defocusing media

### **1. Introduction**

Optical solitons arise as localized optical fields that preserve their shapes during propagation when the medium's diffraction and self-phase modulation are balanced [1]. Since they were first observed experimentally in 2003 [2], spatial solitons [3] in nonlinear optical systems with additional optical lattices have received a significant interest. Currently, fundamental solitons and vortices on a variety of lattices including real and complex structures in the media with cubic (Kerr) [4, 5], saturable [6] and with competing nonlinearities [7] have been studied. Solitons may also exist in aperiodic or quasicrystal lattices [8–12] and in lattices with defects [13] and dislocations [14, 15].

In the studies above, the governing equations are nonlinear cubic Schrödinger (NLS) equation and their variants, e.g., NLS with additional terms and/or with external lattices. On the other hand, numerous optical materials, such as potassium niobate (KNbO3) [16] or lithium niobate (LiNbO3) [17], acquire quadratic nonlinear responses as well [18–24]. NLS equation with coupling to a mean term (NLSM) system governs the pulse dynamics in quadratically polarized media. The general NLSM system is given by [18, 25, 26].

$$\left| i u\_x + \Delta u + |u|^2 u - \rho u \phi = 0, \quad \phi\_{xx} + \nu \phi\_{yy} = \left( |u|^2 \right)\_{xx} \tag{1}$$

Here *u x*ð Þ , *y* is the normalized amplitude of the envelope of the static electric field propagating in the *z* direction. Δ*u* � *uxx* þ *uyy* corresponds to diffraction, and the nonlinear cubic term comes from the nonlinear Kerr effect. The parameter *ρ* is a coupling constant that emerges from the combined optical rectification and electro-optic effects formed by the *ϕ*ð Þ *x*, *y* field, and *ν* is the coefficient that originates from the anisotropy of the material [26]. These equations arise due to the interaction between the fundamental and dc fields when second-harmonic-generation is not phase-matched. In such a situation, the second harmonic component can be solved explicitly and generates an additional self-phase modulation contribution due to cascaded nonlinearity. Consequently, the NLSM system is obtained as a result of the nonlocal nonlinear coupling between the first field (with the cascaded effect from the second harmonic) and a static field that arises from the zeroth harmonic (mean term). We would like to note that solitons of a media with solely quadratic nonlinear response can also be governed by the NLSM type Equations [25, 26].

NLSM equations were derived by Benney and Roskes [27] for a finite water depth, neglecting the surface tension. In 1974, Davey and Stewartson [28] investigated the evolution of a three-dimensional wave packet for a finite water depth and obtained an equivalent form of these equations. The integrability of the NLSM equations that were derived for the shallow water limit was studied by Ablowitz and Haberman [29]. Djordevic and Reddekopp [30] improved the study of Benney and Roskes by including the surface tension in 1977. Later, Ablowitz et al. [18, 25, 26] derived from first principles that NLSM type equations model the evolution of the electromagnetic field in a quadratically polarized media.

One major drawback of the dynamics of solitons under the NLS equation in 2D is that they exhibit collapse. As a matter of fact, in [31], it was revealed that the collapse dynamics of NLSM solitons are similar to the collapse of NLS solitons. Possible collapse arrest mechanisms have been studied extensively in nonlinear optics, e.g., nonlinear saturation [32, 33]. Merle and Raphael [34] investigated the collapse of the NLS solitons and its variants in depth. Furthermore, Gaeta and coworkers [35] carried out detailed experiments in order to expose the nature of the singularity formation in cubic optical media and demonstrated experimentally that collapse occurs with a self-similar profile.

Recently, wave collapse in the NLSM system was studied in [36] and it is shown that in both water waves and optics, collapse occurs with a quasi self-similar profile. By evolving the NLSM solutions that are computed by the Spectral Renormalization method which is essentially a fixed point iteration scheme. In this study, it is also revealed that the NLSM solitons have astigmatic profiles and their collapse can be arrested by adding nonlinear saturation into the system. Wave self-rectification and beam ellipticity as a collapse arrest mechanism for NLSM solitons was put forward by in [16]. In the aforementioned work, for simplicity, the authors considered Gaussian profiles with various input powers and astigmatism and then evolve these profiles for long distances. More recently, wave collapse in the NLSM system was arrested by adding a periodic external lattice in [23] and it has been shown that deeper lattices may serve as a collapse arrest mechanism.

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

In this chapter, the NLSM model (1) is extended by adding a quintic term and an external lattice. Using this new model, the dynamics of the fundamental lattice solitons in a nonlocal nonlinear medium with quintic nonlinearity are explored. The study is concentrated on effects formed by the imposed lattice depth and the strength of quintic nonlinearity. The stability of the obtained lattice solitons is examined by the nonlinear evolution and linear spectra. In the light of the conducted numerical analysis, it is confirmed that fundamental lattice solitons can exist in nonlocal nonlinear media with both self-focusing and self-defocusing quintic nonlinearities and stability of these solitons is achieved for a broad range of system parameters. Since many optical materials such as chalcogenide glasses are engineered to reveal fifth and seventh-order effects in addition to cubic nonlinear effects [37], and high-order nonlinearities can arise even with pure Kerr materials [38–40], it is crucial to consider the soliton dynamics consists of higher-order nonlinearities.

The outline of the chapter is as follows: In Section 2, we present the model equations and compute the periodic lattice solitons numerically. In Section 3, we explore the linear spectra and nonlinear evolution of the solitons. Results are outlined in Section 4.

### **2. The model and its soliton solutions**

It is known that, the steady-state solutions of the NLSM system (1) collapse [36], and the collapse of solitons can be arrested by adding external lattices [23]. The NLSM model (1) is extended as follows to describe the nonlocal qubic-quintic nonlinear medium with an external lattice,

$$\dot{u}u\_x + \frac{1}{2}\Delta u + \beta |u|^2 u - \rho u \phi + \gamma |u|^4 u - V(x, y)u = 0, \qquad \phi\_{xx} + \nu \phi\_{yy} = \left( |u|^2 \right)\_{xx} \tag{2}$$

where *γ* denotes the strength of quintic nonlinearity and *V x*ð , *y*) shows the optical lattice. We consider lattices that are created by the sum of *N* phasemodulated plane waves [8].

$$V(\mathbf{x}, \boldsymbol{\upchi}) = \frac{V\_0}{N^2} \left| \sum\_{n=0}^{N-1} e^{i \left(k\_x^\* \mathbf{x} + k\_\boldsymbol{\upchi}^\* \boldsymbol{\upchi} \right)} \right|^2 \tag{3}$$

where *V*0 >0 is the depth of lattice and the wave vector *kn x*, *kn y* � � ¼

½ � *Kcos*ð Þ 2*πn=N* ,*Ksin*ð Þ 2*πn=N* . The lattices for *N* ¼ 2, 3, 4, 6 correspond to crystal (periodic) structures, while *N* ¼ 5, 7 correspond to aperiodic (Penrose) quasicrystals. Contour plot and diagonal cross-section of the lattice *V x*ð Þ , *y* is displayed in **Figure 1** for *V*0 ¼ 12*:*5, *N* ¼ 4 and *kx* ¼ *ky* ¼ 2*π*. It can be seen that the lattice is periodic, and the center (ð Þ¼ *x*, *y* ð Þ 0, 0 ) of lattice is a local maximum.

### **2.1 Numerical solution for the fundamental solitons**

The squared operator (SOM) method (that was proposed by Yang and Lakoba [41]) is utilized to calculate the soliton solutions of the (2 + 1)D NLSM model (2). The SOM method is based on integrating squared-operators of evolution equations. Derivation of these operators and the scheme of the method are explained below.

**Figure 1.** *The top view and diagonal cross-section of the lattice V x*ð Þ , *y when V*0 ¼ 12*:*5, *N* ¼ 4 *and x*ð Þ , *y* ∈ ½ � �15, 15 *.*

A real-valued nonlinear evolution equation can be presented in the following form

$$\mathbf{L}\_0 \mathbf{u}(\mathbf{x}) = \mathbf{0} \tag{4}$$

where **x** is the multidimensional spatial variable, **u x**ð Þ is a real valued function and the operator **L**0 includes the solitary wave's propagation constant. Let the operator **L**1 denotes linearization of Eq. (4) around the solution **u**, and given by

$$\mathbf{L}\_1(\mathbf{u} + \tilde{\mathbf{u}}) = \mathbf{L}\_1 \tilde{\mathbf{u}} + O\left(\tilde{\mathbf{u}}^2\right) \tag{5}$$

where *u*~ ≪ 1. In order to get a solitary wave solution, the following timedependent squared operator evolution equation will be integrated

$$\mathbf{u}\_t = -\mathbf{M}^{-1} \mathbf{L}\_1 \mathbf{^\dagger M}^{-1} \mathbf{L}\_0 \mathbf{u} \tag{6}$$

here **L1** † denotes the Hermitian of the operator and M is a real valued positive definite Hermitian operator that is introduced to accelerate the convergence. Using the forward Euler method, the solitary wave solution **u** will be calculated by the following iteration.

$$\mathbf{u\_{n+1}} = \mathbf{u\_n} - \left[\mathbf{M^{-1}L\_1{^\dagger}\mathbf{M^{-1}L\_0}u}\right]\_{\mathbf{u=u\_n}}\Delta t. \tag{7}$$

It was shown that the SOM method converges to a solitary wave solution for a broad-range of nonlinear evolution equations when a convenient initial condition is given and time step Δ*t* is small enough [41, 42].

To calculate a solitary wave solution of the model Eq. (2) by the SOM, the following scheme is constructed. Inserting the solution suggestion *u* ¼ *U x*ð Þ , *y* exp ð Þ *iμz* into the model (2), the following sub-operators are obtained

$$\begin{aligned} F\_0 &= -\mu + \beta \left| U \right|^2 - \rho \phi + \gamma \left| U \right|^4 - V(\mathbf{x}, y), \qquad \phi\_{\text{xx}} + \nu \phi\_{\text{yy}} = \left( \left| U \right|^2 \right)\_{\text{xx}} \\ F\_1 &= -\mu + 3\beta \mathcal{U}^2 - \rho \phi + 5\gamma \mathcal{U}^4 - V(\mathbf{x}, y) \end{aligned} \tag{8}$$

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

where *μ* is propagation constant. Using *F*0 and *F*1, we get **L**0 and **L**1 operators as follows:

$$\begin{aligned} \mathbf{L}\_0 U &= \frac{1}{2} \Delta U + F\_0 U, \qquad M\_0 = \mathcal{F}^{-1} \left( \frac{\mathcal{F}(\mathbf{L}\_0 U)}{K^2 + c} \right), \\ \mathbf{L}\_1 U &= \frac{1}{2} \Delta M\_0 + F\_1 M\_0, \qquad M\_1 = \mathcal{F}^{-1} \left( \frac{\mathcal{F}(\mathbf{L}\_1 U)}{K^2 + c} \right) \end{aligned} \tag{9}$$

where

$$\mathcal{F}\{\left|f(\mathbf{x},\boldsymbol{y})\right\rangle\}=\hat{f}\left(\mathbf{k}\_{\mathbf{x}},\mathbf{k}\_{\mathbf{y}}\right)=\int\_{-\infty}^{\infty}f(\mathbf{x},\boldsymbol{y})e^{i\left(\mathbf{k}\_{\mathbf{x}}\mathbf{x}+\mathbf{k}\_{\mathbf{y}}\mathbf{y}\right)}\,d\mathbf{x}d\mathbf{y}\tag{10}$$

$$\mathcal{F}^{-1}\left\{\hat{f}\left(k\_{\mathbf{x}},k\_{\mathcal{Y}}\right)\right\} = f(\mathbf{x},\boldsymbol{y}) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} \hat{f}\left(k\_{\mathbf{x}},k\_{\mathcal{Y}}\right) e^{-i\left(k\_{\mathbf{x}}\mathbf{x}+k\_{\mathcal{Y}}\right)} \,d k\_{\mathbf{x}} d k\_{\mathcal{Y}},\tag{11}$$

*K*2 ¼ *k*2 *x* þ *k*2 *y* and the parameter *c* that has a considerable effect on convergence of the SOM method is utilized for parametrizing the numerical procedure.

After obtaining *M*1, the iteration scheme is executed as follows,

$$U\_{n+1} = U\_n - M\_1 \Delta t.\tag{12}$$

while the mean term *ϕ*ð Þ *x*, *y* is calculated by

$$\phi\_n = \mathcal{F}^{-1}\left(\frac{k\_x^2}{k\_x^2 + \nu k\_y^2} \mathcal{F}\left(|U\_n|^2\right)\right). \tag{13}$$

In order to avoid division by zero error, the first element of *K*2 is set to be 1. Starting from an initial guess, this numerical scheme is iterated until the error *E* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∥*Un*þ1 � *Un*∥2 p <10�8. It is noted that *c* and Δ*t* are chosen heuristically as positive real numbers to obtain a convergent solution, and we take *c* ¼ 1*:*3 and Δ*t* ¼ 0*:*2 in this chapter. The initial condition of the SOM method is chosen as a Gaussian *u x*ð Þ¼ , *y* exp � ð Þ *x* � *x*0 2 þ *y* � *y*0 � �2 Þ � � h i . The location of the initial condition is determined by *x*0 and *y*0. In previous studies [8, 15, 23], it was shown that the solitons centered at the maxima of the lattices cannot stay stable. Thus, in this study, the initial condition is located on a local minimum of the periodic lattice where *x*0, *y*0 � � ¼ ð Þ *π*, 0 (that is shown in **Figure 1**). Unless otherwise stated, the parameters in the model (2) are fixed to the following values

$$(\mu, \rho, \nu, \beta, \gamma, V\_0) \, = \, (-0.1, 0.5, 1.5, 2, \pm 0.2, 12.5). \tag{14}$$

It should be noted that *ρ* ¼ 0*:*5 and *ν* ¼ 1*:*5 are specifically chosen to characterize the electro-optical effects in potassium niobate (KNbO3) [16], and *γ* ¼ 0*:*2 and *γ* ¼ �0*:*2 cases are chosen to investigate soliton dynamics with self-focusing and selfdefocusing quintic nonlinearity, respectively.

The fundamental solitons of the NLSM model (2) are obtained by the SOM method for the considered parameters (14) and are shown in **Figure 2**.

As shown in previous studies [23, 24, 36], fundamental solitons of the NLSM system (2) (with or without lattice) are not radially symmetric due to the anisotropy in the medium. To investigate the effect of the quintic nonlinearity on the level

**Figure 2.**

*3D profiles (first column), top views (second column) of fundamental solitons that are obtained (a) for γ* ¼ �0*:*2 *(self-defocusing) and (b) for γ* ¼ 0*:*2 *(self-focusing) quintic nonlinearity. All other parameters are taken as in Eq. (14).*

of astigmatism in the solitons, the following formulation is defined as a measure of astigmatism

$$\varepsilon = \frac{\text{radius along } y - \text{axis}}{\text{radius along } x - \text{axis}}. \tag{15}$$

Here, *e* ¼ 1 when the soliton is radially symmetric, and if *e*<1 and *e*>1 the obtained solitons are comparatively wider along the *x* and *y* axes, respectively. Thus, the solitons are elliptical if *e* 6¼ 1.

Contour plots of solitons are depicted in **Figure 3** for *ρ* ¼ 0, *ρ* ¼ 0*:*5 and *ρ* ¼ 1 when the quintic nonlinearity coefficient *γ* ¼ �0*:*5, *γ* ¼ 0 and *γ* ¼ 0*:*5. **Figure 3** shows that astigmatism along the *x*-axis is emphasized when the fundamental solitons are obtained with a larger coupling parameter *ρ* (see the first column). It is also observed that the fundamental soliton becomes less astigmatic along *x*-axis with a larger quintic nonlinearity parameter *γ* (for a fixed *ρ*), and when *ρ* ¼ 0 and the quintic nonlinearity is self-focusing (i.e., when *γ* > 0) the solitons become comparatively wider along *y*-axis (see the first row).

To see the impact of the anisotropy parameter *ν* on the soliton profile, the onaxes amplitudes and contour images of fundamental solitons are displayed in **Figure 4**. **Figure 4** shows that the amplitude of fundamental solitons are decreasing with increasing *ν* (from 0 to 3), and solitons become more extended along *x*-axis with larger *ν* values.

Comparing the contour images in **Figures 3** and **4**, it is seen that the astigmatism of the fundamental solitons changes significantly with *ρ*, whereas it depends weakly on *ν*.

### **3. Stability analysis**

The stability properties of fundamental solitons (that are obtained by the SOM method) are examined by the linear eigenvalue spectra and the direct simulation of the nonlinear model (2).

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

**Figure 3.**

*Contour image of fundamental solitons that are obtained for ρ* ¼ 0 *(the first row), ρ* ¼ 0*:*5 *(the second row) and ρ* ¼ 1 *(the third row) while γ* ¼ �0*:*5 *(the first column), γ* ¼ 0 *(the second column) and γ* ¼ 0*:*5 *(the third column). All other parameters are taken as in Eq. (14) and e shows the degree of astigmatism for each soliton.*

The linear stability of solitons are studied by the linear spectra of fundamental solitons. To obtain the linear spectrum, the fundamental soliton is perturbed as follows

$$U = e^{i\mu x} \left[ u\_0(\varkappa, \jmath) + \mathbf{g}(\varkappa, \jmath) e^{i\varkappa} + h^\*(\varkappa, \jmath) e^{i^\*x} \right] \tag{16}$$

where *u*0ð Þ *x*, *y* is the fundamental soliton and *g*, *h* ≪ 1 are infinitesimal perturbations. Substituting the solution *U* into (2) and linearizing, it is found that these normal modes satisfy the following linear eigenvalue problem

$$
\mathcal{L}\mathbf{V} = \mathcal{X}\mathbf{V} \tag{17}
$$

where

$$\mathcal{L} = i \begin{pmatrix} \mathbf{0} & \mathcal{L}\_{+}\\ \mathcal{L}\_{-} & \mathbf{0} \end{pmatrix}, \qquad \mathbf{V} = \begin{pmatrix} \mathbf{g} \\ h \end{pmatrix}.$$

and the matrix coefficients of L are

$$\begin{aligned} \mathcal{L}\_{+} &= -\mu + \frac{1}{2}\Delta - \rho\phi + 3\rho u^2 + 5\chi u^4 - V, \\ \mathcal{L}\_{-} &= -\mu + \frac{1}{2}\Delta - \rho\phi + \beta u^2 + \chi u^4 - V. \end{aligned} \tag{18}$$

**Figure 4.**

*The on-axes amplitudes of fundamental solitons along x-axis (first column), along y-axis (second column) and contour image of solitons (third row) (a) for ν* ¼ 0*; (b) for ν* ¼ 1*:*5 *and (c) ν* ¼ 3*. All other parameters are taken as in Eq. (14) and e shows the degree of astigmatism for each soliton.*

We can solve this eigenvalue problem by the Fourier collocation method [42]. Any eigenvalue with a positive real part in the spectrum indicates the linear instability of the fundamental soliton considered.

The power (or energy is some other contexts) of solitons can be computed by P=ÐÐ �∞j j *u* 2 *dxdy*. In [43], Vakhitov and Kolokolov (VK) demonstrated that there is a critical power value above which the solitons cannot be linearly stable, and a soliton can be linearly stable only if its power increases as the propagation constant *μ* is increased, i.e.,

$$dP/d\mu > 0.\tag{19}$$

Moreover, in [44, 45], it was shown that the solitons can be stable nonlinearly only if the slope condition (19) is satisfied.

In the light of these results, prior to the nonlinear stability analysis, the power and linear stability of solitons are studied in **Figure 5**. These power-eigenvalue diagrams are calculated for the parameters that are given in Eq. (14), and the linear (in)stability intervals of gap solitons are determined via the computation of eigenvalue spectra of the solitons at each point of the power curves (see **Figure 5 (a1)**–**(b1)**). Furthermore, the maximum real part in the eigenvalue spectra (spectral radius) is examined in **Figure 5(a2)**–**(b2)** for the gap solitons. The solitons are found to be linearly stable for self-defocusing quintic nonlinearity (*γ* ¼ �0*:*2) when the power *P*∈ ½ � 1*:*61, 2*:*39 and propagation constant *μ*∈½ � �0*:*35, 0

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

**Figure 5.**

*The power-eigenvalue (P* � *μ) diagram of lattice solitons (a1) for γ* ¼ �0*:*2 *(self-defocusing) and (b1) for γ* ¼ 0*:*2 *(self-focusing). Maximum real part (spectral radius) of the eigenvalue spectra of solitons (a2) for γ* ¼ �0*:*2 *and (b2) for γ* ¼ 0*:*2*. Solid blue and red dotted lines show stable and unstable regions for the gap solitons, respectively.*

(see **Figure 5(a1)**–**(a2)**), and the solitons are linearly stable when *P*∈ ½ � 1*:*76, 2*:*17 and *μ*∈ ½ � �0*:*15, 0*:*3 for self-focusing (*γ* ¼ 0*:*2) quintic nonlinearity (see **Figure 5 (b1)**–**(b2)**). These results are consistent with VK stability criterion since the slope of both power-eigenvalue diagrams is positive.

In addition to the power-eigenvalue diagram, the power and linear stability of solitons are studied in **Figure 6** for varied values of *γ*, *ρ* and *β*. The *x*-axis of each panel shows the existence domain for the varied parameter when other parameters are fixed to the values in Eq. (14). For instance, when *μ* ¼ �0*:*1, *ρ* ¼ 0*:*5, *ν* ¼ 1*:*5, *β* ¼ 2 and *V*0 ¼ 12*:*5, fundamental solitons can be generated for *γ* ∈½ � �0*:*7, 25*:*1 (see the left panel in **Figure 6(a)**). In **Figure 6(a)**, stable (solid blue line) and unstable (red dotted line) regions are determined in the left panel, and in the right panel zoom-in view of the black rectangular region in the left panel is depicted.

**Figure 6** shows that the power of solitons is decreased as *γ* and *β* are increased. The right panel in **Figure 6(a)** shows that solitons are linearly stable when *γ* is between �0*:*24 and 0*:*26, and comparing the left and right panels in **Figure 6(b)** and **(c)**, it is seen that stability regions of *ρ* and *β* parameters for self-focusing (*γ* ¼ 0*:*2) case of quintic nonlinearity is narrow than that of self-defocusing (*γ* ¼ �0*:*2) case.

To confirm obtained linear stability results, we study nonlinear evolution of solitons by means of direct simulation of the governing Eq. (2). To this end, using the fourth-order Runge–Kutta (RK4) scheme fundamental soliton is advanced in *z* direction, and the spatial domain ð Þ *x*, *y* is discretized by the finite-difference method. It should be noted that the RK4 method is widely used for numerical analysis of nonlinear evolution equations, in this regard we apply RK4 to compare the results of this chapter with previous studies. The starting point of nonlinear evolution is chosen as the fundamental soliton, and 1% random noise is inserted to amplitude of the soliton to test the nonlinear stability under perturbations.

### **Figure 6.**

*(a) Soliton power for varying quintic nonlinearity coefficient γ, left panel shows the domain of existence, and right panel shows zoom-in view of the black rectangular region. Soliton powers for γ* ¼ �0*:*2 *(left panel) and γ* ¼ 0*:*2 *(right panel) (b) with varying ρ and (c) with varying β parameters. The linear stability and instability regions are shown by solid blue and red dotted lines, respectively.*

In **Figure 7**, linear stability spectrum (first column), nonlinear evolution of the peak amplitude (second column), the evolution of soliton power (third column), and 3D view of the evolved soliton (fourth column) are plotted for the fundamental solitons that correspond to "a", "b", "c", "d" and "e" points in the right panel of **Figure 6(a)**. Here, the impact of quintic nonlinearity is examined by increasing *γ* from �0*:*3 to 0*:*3.

In **Figure 7(b)**–**(d)**, it is shown that the linear spectrum of the fundamental soliton is purely-imaginary (none of their eigenvalues have a real part), and the peak amplitudes oscillate during the propagation, thus stable evolution can be achieved for the considered solitons in the quadratic-cubic-quintic nonlinear medium when *γ* ¼ �0*:*2, *γ* ¼ 0 and *γ* ¼ 0*:*2. On the other hand, when the coefficient of quintic nonlinearity is increased, there are eigenvalues with positive real part in linear stability spectra, and the peak amplitudes blow-up after a short propagation distance for *γ* ¼ �0*:*3 and *γ* ¼ 0*:*3 as shown in **Figure 7(a)** and **(e)**, respectively.

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

### **Figure 7.**

*Linear spectra (first column), nonlinear evolution of peak amplitudes (second column), the evolution of soliton power (third column), and 3D view of the soliton after evolution (fourth column) for (a) γ* ¼ �0*:*3*; (b) γ* ¼ �0*:*2*; (c) γ* ¼ 0*; (d) γ* ¼ 0*:*2*; and (e) γ* ¼ 0*:*3*. All other parameters are fixed to the values in Eq. (14).*

This fact reveals that both strong self-focusing and self-defocusing quintic nonlinearities have an adverse effect on the stability of fundamental solitons.

From previous studies, it is known that increasing potential depth *V*0 improves the stability of solitons in quadratic [23] and cubic (Kerr) medium [4]. In **Figure 8**, to examine the impact of the deeper lattice on the soliton stability in the quadraticcubic-quintic nonlinear medium when *γ* ¼ �0*:*3 (see panel (a)) and *γ* ¼ 0*:*3 (see panel (b)), the evolution of the peak amplitudes are displayed for varied potential depths. To compare the propagation distance, the amplitude of solitons are divided by their peak amplitudes. **Figure 8** shows that larger lattice depth *V*0 supports the stability of solitons for *γ* ¼ 0*:*3 (self-focusing), whereas shallow potential extends and eventually stabilizes the soliton for *γ* ¼ �0*:*3 (self-defocusing). Thus, when the instability emerges from strong self-focusing nonlinearity, a deeper lattice can be utilized to arrest (or delay) the collapse. This result is consistent with the previous studies, and it is meaningful because the potential in our model (2) is defocusing and it must be balanced with a self-focusing term to obtain stable modes.

Furthermore, it is known that increasing anisotropy parameter *ν* assists the stability of solitons when quadratic electro-optic effects are strong [23, 46, 47]. In

### **Figure 8.**

*Peak amplitude of the evolved solitons for varying lattice depths when the fundamental soliton is obtained for (a) γ* ¼ �0*:*3*; and (b) γ* ¼ 0*:*3*.*

order to investigate the impact of anisotropy on soliton dynamics, we plot the evolution of peak amplitude when *ν* is varying between 0 and 1000 in **Figure 9**. As shown in **Figure 9**, increasing the anisotropy coefficient *ν* (from 0.001 to 100) extends the propagation distance of the evolved solitons, and it stabilizes the soliton in the medium with a strong self-defocusing nonlinearity (when *γ* ¼ �0*:*3) eventually (see black dotted line in panel (a)). Similarly, increased *ν* supports the stability of the solitons for strong self-focusing nonlinearity (when *γ* ¼ 0*:*3), and although it cannot stabilize the soliton robustly, it provides a significant extension of propagation distance when *ν* ¼ 1000 (see black dotted line in panel (b)).

It must be noted that even though soliton solutions of the NLSM system (2) can be obtained when *ν*∈ð Þ 0, ∞ and stability of solitons are improved by higher values of *ν*, they cannot be utilized in practical optical applications, due to the fact that *ρ* and *ν* are predetermined constants associated with the type of optical materials.

### **4. Conclusions**

Fundamental lattice solitons were obtained in a nonlocal nonlinear medium with self-focusing and self-defocusing quintic nonlinearity. The steady-state solutions are calculated by the SOM algorithm and the stability properties of solitons have been explored by linear spectra and nonlinear evolution of the amplitude by direct simulation of the nonlinear model. The band-gap boundaries and stability intervals

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

**Figure 9.** *Peak amplitude of the evolved solitons for varying anisotropy coefficients (ν) between 0.001 and 1000 when the fundamental soliton is obtained for (a) γ* ¼ �0*:*3*; and (b) γ* ¼ 0*:*3*.*

of solitons were found and the power of solitons was investigated for varied *ρ*, *β* and *γ* parameters. It has been seen that the power of fundamental solitons is decreased by increasing cubic (*β*) and quintic (*γ*) nonlinearity, while the power is increasing as the quadratic nonlinear response (*ρ*) is increased, and stability regions of *ρ* and *β* parameters for self-focusing (*γ* ¼ 0*:*2) case of quintic nonlinearity is narrow than that of self-defocusing (*γ* ¼ �0*:*2) case.

The stability analysis showed that fundamental lattice solitons can stay stable for the weak self-focusing and self-defocusing quintic nonlinearity. Strong selffocusing (*γ* ¼ 0*:*3) and self-defocusing (*γ* ¼ �0*:*3) quintic nonlinearities shortened the propagation distance of evolved solitons.

Furthermore, it has been observed that when the quintic nonlinearity is strong in the medium, solitons collapse, and the collapse of solitons can be arrested by a deeper lattice for self-focusing quintic nonlinearity (*γ* ¼ 0*:*3), whereas unstable solitons can be stabilized by shallow lattice for strong self-defocusing quintic nonlinearity (*γ* ¼ �0*:*3).

In conclusion, the numerical existence and stability of fundamental lattice solitons have been presented in a nonlocal nonlinear medium with the quintic nonlinear response, and it has been demonstrated that stability of unstable solitons can be improved by modification of potential depth and strong anisotropy coefficient.

### **Conflict of interest**

The authors declare no conflict of interest.

*The Nonlinear Schrödinger Equation*

### **Author details**

Mahmut Bağcı 1 \*†, Theodoros P. Horikis2†, İlkay Bakırtaş 3† and Nalan Antar3†

1 Department of Management Information Systems, Marmara University, Istanbul, Turkey

2 Department of Mathematics, University of Ioannina, Ioannina, Greece

3 Department of Mathematics Engineering, Istanbul Technical University, Istanbul, Turkey

\*Address all correspondence to<EMAIL_ADDRESS>
† These authors contributed equally.

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

### **References**

[1] Stegeman GIA, Christodoulides DN, Segev M. Optical spatial solitons: Historical perspectives. IEEE Journal of Selected Topics in Quantum Electronics. 2000;**6**:1419-1427

[2] Fleischer JW, Segev M, Efremidis NK, Christodoulides DN. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature. 2003;**422**:147-150

[3] Chen Z, Segev M, Christodoulides DN. Optical spatial solitons: Historical overview and recent advances. Reports on Progress in Physics. 2012;**75**:086401

[4] Ablowitz MJ, Antar N, Bakrtaş İ, Ilan B. Band-gap boundaries and fundamental solitons in complex twodimensional nonlinear lattices. Physical Review A. 2010;**81**:033834

[5] Christodoulides DN, Yang J, editors. Parity-Time Symmetry and its Applications. Singapore: Springer; 2018

[6] Göksel I, Bakrtaş I, Antar N. Nonlinear lattice solitons in Saturable media. Applied Mathematics & Information Sciences. 2014;**9**: 377-385

[7] Göksel I, Antar N, Bakrtaş I. Twodimensional solitons in PT-symmetric optical media with competing nonlinearity. Optik. 2018;**156**:470-478

[8] Ablowitz MJ, Ilan B, Schonbrun E, Piestun R. Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures. Physical Review E. 2006;**74**:035601

[9] Ablowitz MJ, Antar N, Bakrtaş İ, Ilan B. Vortex and dipole solitons in complex two-dimensional nonlinear lattices. Physical Review A. 2012;**86**: 033804

[10] Fu Q, Wang P, Huang C, Kartashov YV, Torner L, Konotop VV, et al. Optical soliton formation controlled by angle twisting in photonic moiré lattices. Nature Photon. 2020;**14**: 663-668

[11] Wang P, Zheng Y, Chen X, Huang C, Kartashov YV, Torner L, et al. Localization and delocalization of light in photonic moiré lattices. Nature. 2020; **577**:42-46

[12] Huang C, Ye F, Chen X, Kartashov YV, Konotop VV, Torner L. Localization-delocalization wavepacket transition in Pythagorean aperiodic potentials. Scientific Reports. 2016;**6**: 32546

[13] Bağcı M, Bakrtaş İ, Antar N. Fundamental solitons in parity-time symmetric lattice with a vacancy defect. Optics Communication. 2015;**356**: 472-481

[14] Martin H, Eugenieva ED, Chen Z, Christodoulides DN. Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices. Physical Review Letters. 2004;**92**: 123902

[15] Bağcı M, Bakrtaş İ, Antar N. Vortex and dipole solitons in lattices possessing defects and dislocations. Optics Communication. 2014;**331**: 204-218

[16] Crasovan LC, Torres JP, Mihalache D, Torner L. Arresting wave collapse by wave self-rectification. Physical Review Letters. 2003;**91**: 063904

[17] Schiek R, Pertsch T. Absolute measurement of the quadratic nonlinear susceptibility of lithium niobate in waveguides. Optical Materials Express. 2012;**2**:126-139

[18] Ablowitz MJ, Biondini G, Blair S. Localized multi-dimensional optical pulses in non-resonant quadratic materials. Mathematics and Computers in Simulation. 2001;**56**: 511-519

[19] Hayata K, Koshiba M. Multidimensional solitons in quadratic nonlinear media. Physical Review Letters. 1993;**71**:3275-3278

[20] Torruellas WE, Wang Z, Hagan DJ, VanStryland EW, Stegeman GI, Torner L, et al. Observation of twodimensional spatial solitary waves in a quadratic medium. Physical Review Letters. 1995;**74**:5036-5039

[21] Torner L, Sukhorukov AP. Quadratic Solitons. Optics & Photonics News. 2002;**13**:42-47

[22] Buryak AV, Trapani PD, Skryabin DV, Trillo S. Optical solitons due to quadratic nonlinearities: From basic physics to futuristic applications. Physics Reports. 2002;**370**: 63-235

[23] Bağcı M, Bakırtaş İ, Antar N. Lattice solitons in nonlinear Schrödinger equation with coupling-to-a-meanterm. Optics Communication. 2017;**383**: 330-340

[24] Bağcı M, Kutz JN. Spatiotemporal mode locking in quadratic nonlinear media. Physical Review E. 2020;**102**: 022205

[25] Ablowitz MJ, Biondini G, Blair S. Multi-dimensional pulse propagation in non-resonant *χ*ð Þ2 materials. Physics Letters A. 1997;**236**:520-524

[26] Ablowitz MJ, Biondini G, Blair S. Nonlinear Schrödinger equations with mean terms in nonresonant multidimensional quadratic materials. Physical Review E. 2001;**63**: 046605

[27] Benney DJ, Roskes GJ. Wave instabilities. Studies in Applied Mathematics. 1969;**48**:377-385

[28] Davey A, Stewartson K. On threedimensional packets of surface waves. Proceedings of the Royal Society A. 1974;**338**:101-110

[29] Ablowitz MJ, Haberman R. Nonlinear evolution equations—Two and three dimensions. Physical Review Letters. 1975;**35**:1185-1188

[30] Djordjevic VD, Redekopp LG. On two-dimensional packets of capillarygravity waves. Journal of Fluid Mechanics. 1977;**79**:703-714

[31] Papanicolaou G, McLaughlin D, Weinstein M. Focusing singularity for the nonlinear schrodinger equation. In: Fujita H, Lax PD, Strang G, editors. Nonlinear Partial Differential Equations in Applied Science; Proceedings of The U.S.-Japan Seminar, Tokyo, 1982. Vol. 81. North-Holland Mathematics Studies. North-Holland; 1983. pp. 253-257. DOI: 10.1016/S0304-0208(08)72095-4

[32] Ghidaglia JM, Saut JC. On the initial value problem for the Davey-Stewartson systems. Nonlinearity. 1990; **3**:475-506

[33] Fibich G, Papanicolaou G. Selffocusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension. SIAM Journal on Applied Mathematics. 1999; **60**:183-240

[34] Merle FH, Raphael P. On universality of blow-up profile for *L*2 critical nonlinear Schrödinger equation. Inventiones Mathematicae. 2004;**156**: 565-672

[35] Moll KD, Gaeta AL, Fibich G. Selfsimilar optical wave collapse: Observation of the Townes profile. Physical Review Letters. 2003;**90**: 203902

*Perspective Chapter: Lattice Solitons in a Nonlocal Nonlinear Medium with Self-Focusing… DOI: http://dx.doi.org/10.5772/intechopen.104824*

[36] Ablowitz MJ, Bakrtaş İ, Ilan B. Wave collapse in a class of nonlocal nonlinear Schrödinger equations. Physica D: Nonlinear Phenomena. 2005; **207**:230-253

[37] Chen YF, Beckwitt K, Wise FW, Aitken BG, Sanghera JS, Aggarwal ID. Measurement of fifth- and seventhorder nonlinearities of glasses. Journal of the Optical Society of America B: Optical Physics. 2006;**23**:347-352

[38] Komarov A, Leblond H, Sanchez F. Quintic complex Ginzburg-Landau model for ring fiber lasers. Physical Review E. 2005;**72**:025604

[39] Mohamadou A, LatchioTiofack CG, Kofané TC. Wave train generation of solitons in systems with higher-order nonlinearities. Physical Review E. 2010; **82**:016601

[40] Azzouzi F, Triki H, Grelu P. Dipole soliton solution for the homogeneous high-order nonlinear Schrödinger equation with cubic-quintic-septic non-Kerr terms. Applied Mathematical Modelling. 2015;**39**:1300-1307

[41] Yang J, Lakoba TI. Universallyconvergent squared-operator iteration methods for solitary waves in general nonlinear wave equations. Studies in Applied Mathematics. 2007;**118**: 153-197

[42] Yang J. Nonlinear Waves in Integrable and Nonintegrable Systems. Philadelphia: SIAM; 2010

[43] Vakhitov NG, Kolokolov AA. Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophysics and Quantum Electronics. 1973;**16**:783-789

[44] Weinstein MI. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM Journal on Mathematical Analysis. 1985;**16**: 472-491

[45] Rose HA, Weinstein MI. On the bound states of the nonlinear schrödinger equation with a linear potential. Physica D: Nonlinear Phenomena. 1988;**30**:207-218

[46] Bağcı M. Soliton dynamics in quadratic nonlinear media with twodimensional Pythagorean aperiodic lattices. Journal of the Optical Society of America B: Optical Physics. 2021;**38**: 1276-1282

[47] Bağcı M. Partially PT -symmetric lattice solitons in quadratic nonlinear media. Physical Review A. 2021;**103**: 023530

### **Chapter 2**

## Soliton Like-Breather Induced by Modulational Instability in a Generalized Nonlinear Schrödinger Equation

*Saïdou Abdoulkary and Alidou Mohamadou*

### **Abstract**

We consider the nonlinear Schrödinger equation modified by a rational nonlinear term. The model appears in various studies often in the context of the Ginzburg-Landau equation. We investigate modulational instability by means of a linear stability analysis and show how the nonlinear terms affect the growth rate. This analytical result is confirmed by a numerical simulation. The latter analysis shows that breather-like solitons are generated from the instability, and the effects of the nonlinear terms are again clearly seen. Moreover, by employing an auxiliaryequation method we obtain kink and anti-kink soliton as analytical solutions. Our theoretical solution is in good agreement with our numerical investigation.

**Keywords:** generalized nonlinear schrödinger, modulational instability, breather like-soliton

### **1. Introduction**

The nonlinear Schrödinger equation (NLSE) is the main equation which governs the propagation of pulses in various fields such as nonlinear optical systems, plasmas, fluid dynamics, Bose-Einstein condensation, and condensed matter physics. It has been shown to govern the evolution of a wave packets in weakly nonlinear and dispersive media and has thus arisen in such diverse fields. One other application of this equation is in pattern formation, where it has been used to model some nonequilibrium pattern forming systems. Most notable is the role it plays in understanding the propagation of electromagnetic waves in glass fibers and other optical waveguides [1].

In this paper we consider a NLS equation with inverse nonlinear terms. Inverse nonlinear term has been introduced for the first time by Malomed and al. [2] which has been later studied by [3, 4] in the case of the Ginzburg-Landau equation.

$$
\dot{u}u\_x + pu\_{xx} + \gamma\_1 \frac{\left|\frac{\partial u}{\partial x}\right|^2}{\left|u\right|^2} u + \gamma\_2 \frac{\mathbf{1}}{\left|u\right|^2} u + \gamma\_3 |u|^2 u = \mathbf{0},\tag{1}
$$

where *u* is a complex amplitude that depends on *z* and *x*, *γi* (*i* ¼ 1, 2, 3) is a real constant and represents the nonlinear coefficient, *p* is a real constant and supposes to be the group-velocity dispersion (GVD) coefficient. Notice that Eq. (1) especially with *γ*1 ¼ *γ*2 ¼ 0 appears in many contemporary work in physics and has been shown to be completely integrable [5] and to admit optical solitons by balancing the GVD and Kerr nonlinearity *γ*3 (the self-focusing interaction and defocusing interaction corresponding to bright and dark solitons, respectively). However, in many applications it contains also some small additional terms which destroy these properties. It describes either the propagation of a continuous wave (CW) beam in a planar waveguide or propagation of an optical pulse inside optical fiber, and show that this equation admits analytical solitary solution and exhibits a modulation instability (MI). This instability leads to spatial or temporal modulation of a constant-intensity plane wave.

Modulational instability is one of the nonlinear wave equations associated to NLSE and appears in many physical systems. It indicates that due to the competition between nonlinearity and the dispersive effects, a small perturbation of the initial plane wave may induce an exponential growth of the wave amplitude, resulting in the carrier-wave breakup into a train of localized waves [6].

The NLSE is also one of the original nonlinear partial differential equations, the study of which has lead to fundamental advances in theoretical physics. The study of NLS was motivated by a large number of theoretical problems ranging from optical pulse propagation in nonlinear fibers to hydrodynamics, condensed matter physics and biophysics. It is now known that NLS is one of the few examples of completely integrable nonlinear partial differential equations [7, 8].

The main objective of this paper is to study MI in a generalized NLSE that includes rational nonlinear terms given by Eq. (1). By means of the linear stability analysis we explicitly investigate the stability condition of a launched plane wave. The results show that the MI gain is strongly dependent on the nonlinear parameters as well as the GVD. Our numerical simulations show that those parameters contribute to the formation and the propagation of the soliton like-breather in the systems. Those parameters can generate either stable or unstable solitons. We also investigate analytical soliton solutions. By employing auxiliary equation method we obtain kink and antikink solutions of Eq. (1).

The rest of the paper is organized as follows. The model is introduced in Section 2, which is followed by the analysis of the MI of the CW solutions in Section 3, direct simulations shown the formation of modulated wave as well as breather like-solitons and their stability in Section 4. Exact analytical kink and antikink soliton solutions are reported in Section 5, and the paper is concluded by Section 6.

### **2. Modulational instability**

The nonlinear Schrodinger Equation Eq. (1) has the simplest solution in the form of a continuous wave as *u* ¼ *u*0 exp *i kx* ð Þ � *ωz* , where *u*0 is a constant and *k* and *ω* are respectively the the wave-number and the angular frequency and satisfy the dispersion relation *ω* � *k*2 *p* þ *γ*1*k*2 þ *γ*2 j j *u*0 2 þ *γ*3j j *u*0 2 ¼ 0. Now we focus our attention on the modulational instability in the system. Therefore, we look at solutions of Eq. (1) in the form of

$$u = u\_0(\mathbf{1} + b) \exp i(k\mathbf{x} - a\mathbf{z}),\tag{2}$$

where *b* represents a small perturbation.

Substituting Eq. (2) into the NLS equation Eq. (1) and linearizing the resulting equations, we obtain a linear equation of *b*.

*Soliton Like-Breather Induced by Modulational Instability in a Generalized Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.100522*

$$i\hbar \mathbf{b}\_x + p\mathbf{b}\_{xx} + 2ik p \mathbf{b}\_x + i\gamma\_1 \mathbf{k} \left(\mathbf{b}\_x^\* - \mathbf{b}\_x\right) - \frac{\chi\_2}{|\mu\_0|^2} (\mathbf{b}^\* + \mathbf{b}) + \chi\_3 |\mu\_0|^2 (\mathbf{b}^\* + \mathbf{b}) = \mathbf{0}, \tag{3}$$

Looking for solutions to this function in the form of plane waves *b* ¼ *b*1 exp *i Kx* ð Þþ � Ω*z b*∗ 2 exp � *i Kx* ð Þ � Ω*z* , we obtain the dispersion relation

$$
\Delta = (2p - \chi\_1)kK \pm K\sqrt{\chi\_1^2 k^2 + p^2 K^2 + 2\frac{\chi\_2}{\left|\mu\_0\right|^2}p - 2\chi\_3 p \left|\mu\_0\right|^2},\tag{4}
$$

where the wave number *K* and the frequency Ω characterize linear properties of the modulation wave. The dispersion relation given by Eq. (4) determine the condition for the stability of a plane wave with a wave number *k* in the system. This is the case as long as Ω is real. This stability condition is explicitly depends on the nonlinear parameters *γ*1, *γ*2, and *γ*3. It shows that the CW plane-wave is absolutely stable only in the case *γ*2 1*k*2 þ *p*2*K*2 þ 2 *γ*2 j j *u*0 2 *p* � 2*γ*3*p u*j j 0 2 <0. That is

$$\left(\chi\_1^2 k^2 + p^2 K^2\right) - \sqrt{\Delta} < \left|\mu\_0\right|^2 < \left(\chi\_1^2 k^2 + p^2 K^2\right) + \sqrt{\Delta},\tag{5}$$

with Δ ¼ *γ*2 1*k*2 þ *p*2*K*2 � �2 þ 16*γ*2*γ*3*p*2.

The modulation instability gain is related to the imaginary part of Ω and is given by

$$\mathbf{g} = \operatorname{Im} \left[ |\mathbf{K}| \sqrt{\gamma\_1^2 k^2 + p^2 K^2 + 2 \frac{\mathcal{V}\_2}{|\boldsymbol{\mu}\_0|^2} p - 2 \gamma\_3 p |\boldsymbol{\mu}\_0|^2} \right],\tag{6}$$

**Figure 1** shows the instability gain as a function of the perturbation wave number *K* for *u*0 ¼ 1 and 1*:*5. The gain exists for both positive and negative values of *K* in the range ∣*K*∣<*K*0 ¼ ð Þ 1*=*2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �2*γ*2 1*k*2 j j *u*0 2 � 4*γ*2*p* þ 4*γ*3*p u*j j 0 4 q *=*ð Þ j*u*0j*p* . The peak gain occurs for *K* ¼ *K*0 and has the value *gmax* ¼ ð Þ 1*=*2 *K*0∣ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*γ*2 1*k*2 þ 4*γ*2*p=*j j *u*0 2 � 4*γ*3*p u*j j 0 2 q ∣. Now let us study the latter gain. We have plotted a qualitative study of its behavior. **Figure 1a** shows that the peak gain increased with the amplitude *u*0 increasing as well as it width. In **Figure 1b** one can see the inverse phenomenon. **Figure 2** shows the evolution of the peak gain as a function of nonlinear parameters *γ*2 and *γ*3. Here we are seeing the

**Figure 1.**

*Gain spectrum g K*ð Þ *of modulation instability as a function of wave number with effect of the background amplitude u*0 ¼ 1 *(dashed line) and* 1*:*5 *(soline) when the GVD is* 0*:*5 *for the left-hand panel (a) with γ*1 ¼ 0*:*1*, γ*2 ¼ 0*:*4*, γ* ¼ 0*:*8 *and the GVD is* �0*:*5 *for the right hand-panel (b) with γ*2 ¼ 0*:*8*, γ*3 ¼ 0*:*01*.*

**Figure 2.**

*Maximum gain spectrum gmax of modulation instability for u*0 ¼ 1 *versus nonlinear parameters γ*2 *and γ*3 *(panel (a)), while in panel (b) we show its contour plot.*

increasing of the peak with the nonlinear terms (see **Figure 2a**). There is a limit cycle where the peak remains constant for certain values of both *γ*2 and *γ*3. This is clearly seen through the contour plot in **Figure 2b**. This aspect is better analyzed in **Figure 3** where the peak gain increased by increasing both *γ*2 and *γ*3 in the left side of top panel (a) as well as the gain width. This is confirm by fixing one nonlinear parameter (*γ*2)

### **Figure 3.**

*Gain spectrum g K*ð Þ *of modulation instability as a function of wave number with effect of the nonlnear parameters γ*2*, γ*3*. In the top panel (a) on left we set γ*2 ¼ 0*:*03*, γ*3 ¼ 0*:*3 *for dashed line, γ*2 ¼ 0*:*3*, γ*3 ¼ 0*:*4 *for the solid line, γ*2 ¼ 0*:*5*, γ*3 ¼ 0*:*6 *for o-line. On right (b) we set γ*2 ¼ 0*:*2*, and γ*3 ¼ 0*:*1, 0*:*3, 0*:*6 *(respectively for dashed, solid and o-line. In botton we got on left panel (c) γ*2 ¼ 0*:*01, 0*:*1, 0*:*2 *(respectively for dashed, solid and o-line) and γ*3 ¼ 0*:*3*. For all the previous panels we got u*0 ¼ 1 *and γ*1 ¼ 0*:*1*. On right panel (d) we consider the same values in (c) when u*0 ¼ 2*.*

### *Soliton Like-Breather Induced by Modulational Instability in a Generalized Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.100522*

when the last one (*γ*3) is varying (see panel (b)). One observe the inverse phenomenon by fixing *γ*3 when *γ*2 increasing (see panel (c)). The last panel (d) is very particular while we are seeing the peak and the width gain are almost constant by varying the nonlinear parameters (*γ*2) when the background amplitude *u*0 ¼ 2.

### **3. The numerical simulation analysis**

Analytical analysis done by linear stability shown the possibility of the formation of modulated waves in the consider system. This prediction can be numerically confirmed. In this way let us launch as initial condition a modulated plane wave: *u*ð Þ¼ 0, *x* 1 þ *ε* cosð Þ *Kx* where fixed boundary conditions are used and the numerical constants used in the calculation are the following: *ε* ¼ 0*:*01, *p* ¼ 0*:*5, *k* ¼ 0, *K* ¼ 0*:*2*π*, *γ*1 ¼ 0*:*1, 0*:*2, 0*:*3 in *γ*2 ¼ 0*:*01, 0*:*1, 0*:*4, 0*:*6 and *γ*3 ¼ 0*:*1, 0*:*3, 0*:*6, 0*:*9 normalized units. The question we are going to answer is the influence of the parameters *γi* on the formation of modulated wave.

From **Figure 4** one can see the formation of bright solitary wave. The left hand top panel shows the generation of a pulse train toward the boundary regions but the intensity is smallest at the center. On right hand panel we can see the bright solitary wave behaves like a breather soliton is forming. This may be a multisoliton quasiperiodic solutions. It can be seen that the breather solutions keep their oscillating shapes, while the wave packets move as periodic solitons along the *x*-axis for certain values of *z*. Those breathers are periodic in the *x* coordinate and aperiodic in the *z* coordinate. There is more generation of breathers in bottom panels (e) and (f). Comparing panel (b) with panels (e) and (f), one can see that in panels (e) and (f), under the influence of the increasing values of the parameter *γ*3, the number of peaks on the same space interval is increasing when ∣*x*∣ goes up even *z*. The breathers have compressed in width and peak, and this is clearly seen through the contour plot figures given by panels (c), (d), (g) and (h). Those phenomenon are certainly caused by increasing of the nonlinear parameter *γ*3 when *γ*2 remains small and constant. We can see the evolution of the peak amplitude of the wave over the z-axis for each case above in **Figure 5**. One can see that in panel (a) the peak amplitude increases gradually and oscillation little beat over the parameter *z*. The oscillation of the peak is increasing when the nonlinear parameter *γ*3 increases and the curve believes sharp. This is perceptible in the rest panels (b, c and d). One can clearly confirms The dynamical process of the spatial pattern formation induced by MI. When *γ*3 increases, the rate of MI increases too and the MI occurs earlier. Another interesting phenomenon is the width of the breather which decreases by increasing the consider nonlinear parameter.

There is more breathers when *γ*2 is negative. **Figure 6** shows the evolution of the typical intensity profile done by numerical simulations. In panel (a) one can see that there are more breathers that appears more stable than the previous one. This analysis is more perceptible in panel (b) where we plot the contour plot of the consider figure. We are seeing both presence of breathers and bright soliton. This means that the consider parameter is strongly responsible of the formation of those solitons. This is more view in **Figure 7** where panels (a) and (b) shown how breathers are broke and the bright soliton takes place and propagate through the system when the wave vector is small than the previous one (0*:*01*π*).

### **4. Exact analytical solutions to the consider stationary model**

We now discuss about the analytical solution to the stationary NLS of Eq. (1). Suppose that

*The evolution of the typical intensity profile done by numerical simulation of Eq. (1) when γ*3 *is varying (*0*:*1*, for (a),* 0*:*3 *for (b),* 0*:*6 *for (e) and* 0*:*8 *for (f)) by fixing γ*1 ¼ 0*:*1 *and γ*2 ¼ 0*:*01*, while panels (c), (d), (g) and (h) show their respective contour plots.*

$$u(z, \mathbf{x}) = V(\mathbf{x}) \exp\left[i\phi(z)\right],\tag{7}$$

is the solution of Eq. (1) where *V* is independent of *z* and *ϕ* the phase. Substituting Eq. (7) into Eq. (1) we obtain two equations for *V* and *ϕ*. The phase equation shows that *ϕ* should be of the form *ϕ*ð Þ¼ *z βz*, where *β* is a constant and *V* equation is

*Soliton Like-Breather Induced by Modulational Instability in a Generalized Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.100522*

**Figure 5.**

*Representation of the maximum amplitude versus z corresponding of each panels (a), (b), (e) and (f) of Figure 4 respectivelly.*

**Figure 6.**

*The influence of negative value of γ*2 *on the evolution of the typical intensity profile done by numerical simulation of Eq. (1) with p* ¼ 1*=*2*, γ*1 ¼ 0*:*1*, γ*2 ¼ �0*:*2*, γ*3 ¼ 0*:*4 *showed by panel (a), while panel (b) shows its contour plot.*

$$-\beta V + pV\_{\infty} + \gamma\_1 \frac{V\_x^2}{V} + \gamma\_2 \frac{1}{V} + \gamma\_3 V^3 = 0,\tag{8}$$

For solving this equation we set *V* ¼ *G*1 2 and then the Eq. (8) yields

$$\frac{1}{4}(\chi\_1 - p)\dot{\mathbf{G}}^2 + \frac{1}{2}p\mathbf{G}\ddot{\mathbf{G}} + \chi\_2\mathbf{G} - \beta\mathbf{G}^2 + \chi\_3\mathbf{G}^3 = \mathbf{0},\tag{9}$$

This is a nonlinear ordinary differential equation which can be solve by the auxiliary equation method.

**Figure 7.**

*Influence of γ*2 *and wave number K on the evolution of the typical intensity profile done by numerical simulation of Eq. (1) when γ*2 ¼ �0*:*03 *in (a) and γ*2 ¼ 0*:*03 *(b), the rest of the parameters are γ*1 ¼ 0*:*1*, γ*3 ¼ 0*:*5 *and K* ¼ 0*:*01*π.*

### **4.1 The auxiliary equation method**

The auxiliary equation method has been defined by [9, 10] while it allows to find more and new multiple solutions for nonlinear partial differential equations. The main steps of using this method is summarized as follows.

For solving equation

$$P(\boldsymbol{\mu}, \boldsymbol{\mu}\_t, \boldsymbol{\mu}\_\mathbf{x}, \boldsymbol{\mu}\_\mathbf{x}, \boldsymbol{\mu}\_\mathbf{x}, \dots) = \mathbf{0},\tag{10}$$

we set *ξ* ¼ *x* þ *ωt* then the nonlinear partial differential equation in two independent variables ð Þ *x*, *t* becomes a nonlinear ordinary differential equation

$$Q\left(u, u', u', u''\right) = 0,\tag{11}$$

We seek for the solutions of Eq. (11) in the following generalized form

$$u(\xi) = \sum\_{i=0}^{2M} a\_i F^i(\xi),\tag{12}$$

in which *ai* (i = 0, 1, 2, … , 2 M) are constants to be determined and *M* ¼ 2. The variable *F*ð Þ*ξ* should satisfy the following variable separated ordinary differential equation

$$F^{\prime 2}(\xi) = aF^2(\xi) + bF^4(\xi) + cF^6(\xi),\tag{13}$$

where *a*, *b*, *c* are parameters to be determined. Substituting Eq. (12) into (11) by taking in account Eq. (13) and equate the coefficients of all powers of *F*ð Þ*ξ* to zero yields a set of algebraic equations for unknowns *a*, *b*, *c*, *ai* (i = 0, 1, … , 2 M) and *ω*. We solve the set of algebraic equations by the use of Maple and substitute the solutions obtained in this step back into (12) so as to obtain the exact traveling wave solutions for Eq. (10).

The solution of Eq. (9), balancing *GG*00 with *G*3 gives *M* ¼ 2. Therefore we may choose

$$G^{\prime 2} = a\_0 + a\_1 F(\xi) + a\_2 F^2(\xi) + a\_3 F^3 + a\_4 F^4,\tag{14}$$

*Soliton Like-Breather Induced by Modulational Instability in a Generalized Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.100522*

**Figure 8.**

*Kink and anti-kink representations of the analytical solutions done by Eq. (16). The following parameter values are used p* ¼ 1*=*2*, γ*1 ¼ �1*:*1*, γ*2 ¼ 0*:*3 *and γ*3 ¼ 0*:*6*.*

where *a*0, *a*1 and *a*2, *a*3, *a*4 are constants to be determined. By applying the defined method we obtained the following exact kink and anti-kink solutions for the stationary NLSE (9).

$$G = \pm \sqrt{a\_0} \tanh\left(\mathbf{x} \sqrt{a}\right),\tag{15}$$

where *a*0 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *γ*2 *γ*3 1 þ 2*p γ*1 r � � and *a* ¼ � *γ*2 *γ*1*a*0 . We must have *γ*2*=γ*1 <0 in order to

ensure that the pulse width ffiffiffi *a* p is real.

Having obtained exact solutions of the stationary NLSE Eq. (9), we will use them together to construct soliton solutions of the NLSE Eq. (1). In this case, the kink-soliton and anti-kink solutions of Eq. (1) can be written in the form

$$
\mu(z,\varkappa) = \pm \sqrt{a\_0} \tanh\left(\varkappa \sqrt{a}\right) \exp i\beta z,\tag{16}
$$

where *β* ¼ 2*a γ*ð Þ 1 þ *p* . **Figure 8** shows the representation of the analytical solution to the stationary NLSE.

### **5. Conclusion**

In the present study a generalized nonlinear Schrödinger equation with particular nonlinearities has been introduced. The model including rational nonlinearity that arise from Malomed model and describes the propagation of nonlinear surface waves on a plasma with a sharp boundary. We explicitly investigated MI gain by means of linear stability analysis. Results reveal that the nonlinear parameters are strongly influences the dynamics of the launched plane wave. We further tested the evolutionary modulate plan wave numerically, which indicates that those parameters allow the formation of breather-like soliton in the system as well as bright soliton. We have investigated analytical kink and anti-kink soliton too.

It would be particularly worthwhile to extend this study to the generalized NLS with time and space modulated nonlinearities and potentials. This could allow more stability and more formation of the breather-like soliton as well as the Akhmediev breather [11], Peregrine rogue wave [12], and Kuznetsov-Ma breather [13, 14] and even high-order rogue waves [15]. MI gain distributions could bring different nonlinear excitation pattern dynamics.

*The Nonlinear Schrödinger Equation*

### **Classification**

PACS numbers: 05.45,Yv, 04.20.Jb, 42.65.Tg

### **Author details**

Saïdou Abdoulkary1 \* and Alidou Mohamadou2,3

1 Département des Sciences Fondamentales, FMIP, University of Maroua, Maroua, Cameroon

2 National Advanced School of Engineering of Maroua, Maroua, Cameroon

3 The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

\*Address all correspondence to<EMAIL_ADDRESS>
© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Soliton Like-Breather Induced by Modulational Instability in a Generalized Nonlinear… DOI: http://dx.doi.org/10.5772/intechopen.100522*

### **References**

[1] G. P. Agrawal, *Nonlinear Fiber Optics* (Academic, California, 2001).

[2] B. A. Malomed, L. Stenflo, J Phys A: Math Gen **24** L1149 (1991).

[3] E. Yomba, T. C. Kofane, Chaos, Solitons and Fractals **17** 847 (2003).

[4] A. Mohamadou et al. Chaos, Solitons and Fractals **27** 914–925 (2006).

[5] V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. **61**, 118 (1971) [Soy. Phys. JETP 34, 62 (1972)].

[6] L.Q. English, M. Sato, A.J. Sievers, Phys. Rev. B **67**, 024403 (2003).

[7] D. Hennig, G. P. Tsironis, Physics Reports **307** (1999) 333-432.

[8] M. J. Ablowitz, P. A. Clarkson, *Solitons, Nonlinear Evolution Equations and Inverse Scattering*, Cambridge Univ. Press, New York, 1991.

[9] Sirendareji, Phys. Lett. A **356** (2006) 124.

[10] S. Abdoulkary et al., Appl. Math. Comput. **233** (2014) 109-115.

[11] N. Akhmediev and V. I. Korneev, Theor. Math. Phys. **69**, 1089 (1986).

[12] D. H. Peregrine, J. Aust. Math. Soc. Ser. B 25, **16** (1983).

[13] E. A. Kuznetsov, Sov. Phys. Dokl. **22**,507(1977).

[14] Y. C. Ma, Stud. Appl. Math. **60**, **43** (1979).

[15] B. L. Guo, L. M. Ling, and Q. P. Liu, Phys. Rev. E **85**, 026607 (2012).

### **Chapter 3**

## A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition Method for the Solutions of the Sasa-Satsuma Equation

*Mir Asma*

### **Abstract**

This chapter will talk about the mathematical as well as numerical aspects of the Sasa-Satsuma equation that is the extended nontrivial version of nonlinear Schrödinger's equation. The exact solution will be found out by the undetermined coefficient method. After that, the Adomian decomposition method is secure numerical simulations of computed analytical solutions. The error plots are given to see the accuracy of the results.

**Keywords:** Sasa-Satsuma equation, solitons, Adomian decomposition method, undetermined coefficient method, telecommunication

### **1. Introduction**

Solitons can be illustrated as special wave packets that are formed as a result of elegant balance among the fiber nonlinearity and dispersion. They have the ability to travel undistorted along trans-continental and trans-oceanic distances. Solitons are narrow pulses with immense peak power and exceptional properties. Mostly, pulses go through by spreading because of group velocity dispersion while propagating through optical fibers. Actually, solitons have the advantage of nonlinear effects that helps to overcome the broadening of pulse with the group velocity dispersion. When the corresponding dispersive effects and nonlinear effects are controlled and get the appropriate shape of the pulse. When these pulses balance compression and broadening and there is no change in the shape of the pulse or there are periodic changes in the shape of the pulse, this phenomenon is called soliton. Solitons are very advantageous for optical communication that they can overcome chromatic dispersion completely. In most Dense Wavelength Division Multiplexing (DWDM) systems, fiber losses are compensated periodically by using fiber amplifiers spaced 60-80 Km apart. Attenuation and higher powers are the indirect properties of solitons that are reimbursed by the optical amplifiers. When solitons and amplifiers are used together, they can assure the very high-bit rate, for the repeater-less data transmission for long distances. This combination can be responsible for the data transmission at a bit rate of 80 Gb/s for 10,000 km and it

was testified in the laboratory. Hasegawa and Tappert in 1973, have discussed the possibilities of propagation of solitons through optical fibers and showed their remarkable stability by numerical computation [1]. Seven years later, Mollenauer, Stolon, and Gordon succeeded in observing soliton propagation experimentally [2].

In this chapter, Sasa-Satsuma equation (SSE) is going to be studied for the sake of optical solitons. SSE is the expansion of nonlinear Schrödinger equation (NLSE). In 1991, Narimasa Sasa and Junkichi Satsuma reported significant results that have a great impact in the field of nonlinear optics and the telecommunication industry [3]. Initially, Sasa and Satsuma displayed a nonlinear complex wave equation that was composed with the aid of inverse scattering transformation [4].

The Sasa-Satsuma equation with the linear temporal evolution is [5]:

$$\left[iq\_t + a(|q|^\eta q)\_{\text{xx}} + b|q|^2 q + i\left[\beta\_3 q\_{\text{xxx}} + \sigma(|q|^2)\_{\text{x}} q + \theta|q|^2 q\_{\text{x}}\right] = \mathbf{0} \tag{1}$$

In (1), *q x*ð Þ , *t* is the dependent variable, *x* and *t* are independents variables and the subscripts serve as partial derivatives. The first term in (1) known as time evolution term, while *a* is the coefficient of nonlinear chromatic dispersion, *b* gives the self-phase modulation with kerr nonlinearity, *β*3 is the coefficient of third-order dispersion, *σ* and *θ* are the coefficients of nonlinear dispersion. Finally, *n* gives the nonlinearity parameter of chromatic dispersion and *n* >0.

### **2. Method of undetermined coefficients**

Method of undetermined coefficients gives the spectrum of soliton solutions. In order to seek the soliton solution of SSE to (1) [6–10]:

$$q(\mathbf{x},t) = P(\mathbf{x},t)e^{i\phi(\mathbf{x},t)} \tag{2}$$

Where *P*(*x*,*t*) is the amplitude segment of the soliton. The phase component is

$$
\phi(\mathbf{x}, t) = -\kappa \mathbf{x} + at + \theta\_0 \tag{3}
$$

Here, *κ*, *ω* and *θ*0 are the soliton frequency, wave number and the phase constant respectively. By substituting (2) into (1) and equating real and imaginary parts leads to

$$\begin{aligned} & (-a\kappa^2(\mathbf{1}+n)^2P^{\mathbf{r}+1} + a\left(\frac{\partial}{\partial \mathbf{x}}P(\mathbf{x},t)\right)^2n(\mathbf{1}+n)P^{\mathbf{r}-1} + (a(\mathbf{1}+n)(P(\mathbf{x},t))^n + 3\beta\_3\kappa)\frac{\partial^2}{\partial \mathbf{x}^2}P(\mathbf{x},t) \\ & \mathbf{P}(\mathbf{x},t)\left(\left((-2\sigma-\theta)\kappa-b\right)(P(\mathbf{x},t))^2 + \beta\_3\kappa^3 + a\right) = \mathbf{0}, \end{aligned} \tag{4}$$

and

$$\begin{cases} \frac{\partial^3}{\partial \mathbf{x}^3} P(\mathbf{x}, t) \Big) \beta\_3 + \left( -2a\kappa \left( 1 + n \right)^2 (P(\mathbf{x}, t))^n + \left( 2\sigma + \theta \right) \left( P(\mathbf{x}, t) \right)^2 - 3\beta\_3 \kappa^2 \right) \frac{\partial}{\partial \mathbf{x}} P(\mathbf{x}, t) \\\\ + \frac{\partial}{\partial t} P(\mathbf{x}, t) = \mathbf{0}. \end{cases} \tag{5}$$

*A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition… DOI: http://dx.doi.org/10.5772/intechopen.101817*

### **2.1 Solution of bright soliton**

For the solution of bright soliton, the starting hypothesis is [7–10]

$$P(\mathbf{x},t) = \frac{A}{(D + \cosh \tau)^p},\tag{6}$$

Where

$$
\pi = B(\mathbf{x} - vt) \tag{7}
$$

*A* is known as amplitude, *B* is the inverse width of the soliton, and *D* is the parameter that relates to A and B. The p is an unknown parameter that can be found with the aid of balancing principle. By putting (6) into (4) gives:

$$\begin{aligned} \frac{A^3(\kappa \theta + b)}{E^{3p}} + \frac{A\left(a\left(n p^2 (n+2) B^2 - \kappa^2\right) + 3B^2 \beta\_3 \kappa p^2 - \beta\_3 \kappa^3 - a\right) + AB^2 p^2 a}{E^p} \\ + \frac{((n+1)(1+(n+1)p)a + 3\beta\_3 \kappa (p+1))(D+1)(D-1)ApB^2}{E^{p+2}} \\ -2\frac{DAp((n+1)(1/2 + (n+1)p)a + 3\kappa (p+1/2)\beta\_3)B^2}{E^{p+1}} = 0. \end{aligned} \tag{8}$$

where *E* ¼ *D* þ ð Þ cosh ð Þ*τ* .

With the balancing principle, the exponents of 3*p* and *p* þ 2 gives *p* ¼ 1. By setting the coefficients of linearly independent function in (8) to zero that gives;

$$
\rho = \left( \left( n + \mathbf{1} \right)^2 \boldsymbol{a} + \mathbf{3} \beta\_3 \kappa \right) \boldsymbol{B}^2 - \beta\_3 \kappa^3 - a \kappa^2 \tag{9}
$$

$$A = \pm \frac{\sqrt{(\kappa \theta + b)(D - 1)(D + 1)((n^2 + 3n + 2)a + 6\beta\_3 \kappa)}B}{\kappa \theta + b},\tag{10}$$

$$a \,\theta + b > 0,$$

$$0 \,\, b > 0,$$

$$D = \frac{\sqrt{\left( (n^2 + 3n + 2)a + 6\beta\_3\kappa \right) \left( -B^2 (n+2)(n+1)a - 6B^2 \beta\_3\kappa + (\kappa\theta + b)A^2 \right)}}{\left( (n^2 + 3n + 2)a + 6\beta\_3\kappa \right)B},$$

$$\text{(12)}$$

$$((n^2 + 3n + 2)a + 6\beta\_3 \kappa)B > 0. \tag{13}$$

The solution of bright soliton of (1) is

$$q(\mathbf{x}, t) = \frac{A}{D + \cosh\left[B(\mathbf{x} - vt)\right]} e^{i(-\kappa \mathbf{x} + a\mathbf{t} + \theta\_0)},\tag{14}$$

**Figure 1** represents the bright optical soliton of SSE. The soliton solution appears with their corresponding constraints conditions.

### **2.2 Solution of dark solitons**

For the solution of dark soliton, the assumption is [7–10]

$$P(\mathbf{x}, t) = (A + B \tanh \tau)^p,\tag{15}$$

**Figure 1.** *Bright soliton.*

where *A* and *B* are free parameters and

$$
\pi = \mu(\mathbf{x} - \boldsymbol{\nu}t),
\tag{16}
$$

Here, *p* can be found with the balancing principle. By substituting (15) into (4), the real part gives:

$$\begin{aligned} &\mu^2(A(A+B)-B(A+B))^4(a(-1+(n+1)p)(n+1)+3\beta\_3k(p-1))p\,E^{p-2} \\ &-6\left(A-B\right)A(a(-1/2+(n+1)p)(n+1)+2\beta\_3k(p-1/2))(A+B)\mu^2p\,E^{p-1} \\ &+((1+2(n+1)p)a(n+1)+3\beta\_3k(2p+1))A\mu^2p\,E^{p+1} \\ &+((1+(n+1)p)a(n+1)+3\beta\_3k(1+p))\mu^2p\,E^{p+2}+B^2(k\theta+b)E^{3p} \\ &+18\left((A^2-1/3\,B^2)p^2\left(1/3\,(n+1)^2a+\beta\_3k\right)\mu^2-1/18B^2\left(\beta\_3k^3+ak^2+w\right)\right)\mathbf{E}^p=0 \end{aligned} \tag{17}$$

where *E* ¼ ð Þ *A* þ *B* tanh ð Þ*τ*

The value of *p* is similar to bright soliton and gives the value of coefficients of linearly independent function as zero that yields to the following relations of soliton parameters.

$$A = \pm B = \pm \frac{\sqrt{(k\theta + b)(an^2 + 3an + 6\beta\_3 k + 2a)}}{k\theta + b},\tag{18}$$

$$(k\theta + b) > 0,\tag{19}$$

$$(an^2 + 3an + 6\beta\_3 k + 2) > 0,\tag{20}$$

$$\mu = \frac{\sqrt{(k\theta + b)(an^2 + 3an + 6\beta\_3 k + 2a)}}{an^2 + 3an + 6\beta\_3 k + 2} \tag{21}$$

$$
\omega = -ak^2 + 4an^2 - k^3 + 8an + 4a + 12k \tag{22}
$$

*A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition… DOI: http://dx.doi.org/10.5772/intechopen.101817*

Hence, the solution of the dark soliton is given as:

$$q(\varkappa, t) = A \{ \mathbf{1} \pm \tanh \left[ \mu(\varkappa - vt) \right] \} e^{i(-\kappa x + \alpha t + \theta\_0)},\tag{23}$$

**Figure 2** represents the dark soliton of SSE. The soliton solution appears with their corresponding constraints conditions.

### **2.3 Solution of singular solitons**

For the solution of singular soliton, the starting assumption is [7–10];

$$P(\mathbf{x}, t) = \frac{A}{(D + \sinh \tau)^p},\tag{24}$$

Here, *A*, *B*, and *D* are the free parameters with the unknown *p*. By putting (25) into (4) gives;

$$\begin{aligned} &\frac{((n+1)(1+(n+1)p)a+3\beta\_3(p+1)k)\text{Ch}^2ApB^2}{E^{p+2}} - \frac{AB^2\text{Shp}(an+3\beta\_3k+a)}{E^{p+1}} \\ &+ \frac{A^3(k\theta+b)}{E^{3p}} - \frac{A\left(\beta\_3k^3+ak^2+w\right)}{E^p} = 0, \end{aligned} \tag{25}$$

where *E* ¼ ð Þ *D* þ sinh ð Þ*τ* . By setting 3*p* ¼ *p* þ 2, we get *p* ¼ 1 and the free parameters are related as

$$A = \pm \sqrt{\frac{an^2 + 3an + 6\beta\_3 k + 2a}{k\theta + b}} B,\tag{26}$$

$$b \, k \theta + b > 0,\tag{27}$$

$$
\rho = \left( a(n+1)^2 + 3\beta\_3 k \right) \mathbf{B}^2 - \beta\_3 \mathbf{k}^3 - a\mathbf{k}^2 \tag{28}
$$

Hence the solution of singular soliton of (1) is as:

$$q(\mathbf{x}, t) = \frac{A}{D + \sinh\left[B(\mathbf{x} - \nu t)\right]} e^{i(-\kappa \mathbf{x} + \alpha t + \theta\_0)},\tag{29}$$

for designated parameters.

### **2.4 Solution of** *W***-shaped solitons**

For the solution of w-shaped soliton, the starting assumption is [7–10];

$$P(\mathbf{x}, t) = (\beta + \rho \text{sech}(\tau))^p,\tag{30}$$

$$
\mathfrak{r} = \mu(\mathfrak{x} - \mathfrak{v}\mathfrak{t}),
\tag{31}
$$

Substituting (30) into (4) gives

$$\begin{aligned} &4\mu^2 p \beta (a(n+1)(1+2(n+1)p)+3\beta\_3 k(2p+1))E^{p+1} \\ &-p(\beta+\rho)\mu^2 ((-1+(n+1)p)a(n+1)+3\beta\_3 k(p-1))\rho^2(\rho-\beta)E^{p-2} \\ &-2p((-1/2+(n+1)p)a(n+1)+3\beta\_3 k(p-1/2))(-2\rho^2+\rho^2)\mu^2 \beta E^{p-1} \\ &+\rho^2 (k\theta+b)E^{3p} + \left(p^2 \left((n+1)^2a+3\beta\_3 k\right)\mu^2 - \beta\_3 k^3 - ak^2 - w\right)\rho^2 \\ &-6\mu^2 \beta^2 p^2 \left((n+1)^2a+3\beta\_3 k\right)E^p \\ &-((1+(n+1)p)a(n+1)+3\beta\_3 k(1+p))p\mu^2 E^{p+2} = 0,\end{aligned} \tag{32}$$

where *E* ¼ ð Þ *β* þ *ρ* sechð Þ*τ*

By the aid of balancing principle, the value of *p* ¼ 1 that gives parameters as;

$$
\beta = \pm \mathbf{1}/2\sqrt{2}\rho,\tag{33}
$$

$$
\mu = \pm \sqrt{\frac{k\theta + b}{an^2 + 3an + 6k\theta\_3 + 2a}} \rho,\tag{34}
$$

$$
\rho = \pm \sqrt{\frac{an^2 + 3an + 6k\beta\_3 + 2a}{k\theta + b}},\tag{35}
$$

and

$$w = -\frac{6\rho^2\mu^2an^2 - \mu^2an^2\rho^2 + 12an\mu^2\rho^2 - 2a\mu^2n\rho^2 + 18\rho^2\mu^2k\rho\_3 + \rho\_3k^3\rho^2 - 3\rho\_3k\mu^2\rho^2 + 6\mu^2a\rho^2 + ak^2\rho^2 - \mu^2a\rho^2}{\rho^2}.\tag{36}$$

Therefore, *W*-shaped soliton solution is given by:

$$q(\mathbf{x}, t) = (\beta + \rho sech(\tau))e^{i(-\kappa x + \alpha t + \theta\_0)},\tag{37}$$

**Figure 3** represents the W-shaped optical soliton of SSE. The soliton solution appears with their corresponding constraints conditions.

*A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition… DOI: http://dx.doi.org/10.5772/intechopen.101817*

### **3. Numerical investigation of soliton solutions**

In this section, the Adomian decomposition method will be implemented. ADM has gained very much popularity in recent times in applied mathematics. This method is very robust, efficient, and effective to grasp a broad range of linear, nonlinear, ordinary or partial differential equations and linear or nonlinear integral equations. This method gives the fast convergence of the solution and has many symbolic advantages.

Geoge Adomian has introduced and developed this method and very well treated it in the literature. A very appreciable amount of work has been explored for the wide range of linear, nonlinear, ordinary differential equations, partial differential equations as well as integral equation [11].

### **3.1 Recapitulation of Adomian decomposition method**

In this section, ADM is used to handle SSE numerically that show the broad spectrum analytically results. This method tackles the problem in a direct way that shows the accuracy of the exact solution of soliton solutions.

$$q(\mathbf{x},t) = q\_1(\mathbf{x},t) + iq\_2(\mathbf{x},t) \tag{38}$$

By substituting (38) into (4) and breaking it down into real and imaginary parts, respectively

$$\begin{aligned} &-u\_{2t} + a \left(u\_1^2 + u\_2^2\right)\_{\text{xx}}^n u\_{1\text{xx}} + u\_1^3 + u\_1 u\_2^2 - \beta\_3 u\_{2\text{xxx}} - \sigma \left(u\_{1\text{x}}^2 u\_2 + u\_{2\text{x}}^2 u\_2\right) \\ &- \theta \left(u\_1^2 u\_{2\text{x}}^2 + u\_2^2 u\_{2\text{x}}\right) = 0, \end{aligned} \tag{39}$$

$$\begin{aligned} & \left( u\_{1t} + a \left( u\_1^2 + u\_2^2 \right)\_{\text{xx}}^n u\_{2\text{xx}} + u\_2^3 + u\_1^2 u\_2 + \beta\_3 u\_{1\text{xxx}} + \sigma \left( u\_{1\text{x}}^2 u\_1 + u\_{2\text{x}}^2 u\_1 \right) \right) \\ & + \theta \left( u\_1^2 u\_{1\text{x}}^2 + u\_2^2 u\_{1\text{x}} \right) = 0, \end{aligned} \tag{40}$$

The solution is decomposed into finite sums of components by decomposition method that is defined as;

$$u\_i(\mathbf{z}, t) = \sum\_{n=0}^{\infty} u\_{i,n}(\mathbf{x}, t), \tag{41}$$

Here, *i*∈ f g 1, 2 . The components *ui*,*n* , *n*≥ 0 and *i* ¼ 1, 2 will be found out recurrently. By using an operator form *Lt* ¼ *∂ ∂t* , Eqs. (39) and (40) becomes

$$-L\_t u\_2(\mathbf{x}, t) + N\_2(u\_1, u\_2) = \mathbf{0} \tag{42}$$

$$L\_t u\_1(\mathbf{x}, t) + N\_1(u\_1, u\_2) = \mathbf{0},\tag{43}$$

Where

$$\begin{aligned} N\_2(u\_1, u\_2) &= a \left( u\_1^2 + u\_2^2 \right)\_{\text{xx}}^n u\_{1\text{xx}} + u\_1^3 + u\_1 u\_2^2 - \beta\_3 u\_{2\text{xxx}} - \sigma \left( u\_{1\text{x}}^2 u\_2 + u\_{2\text{x}}^2 u\_2 \right) \\ - \theta \left( u\_1^2 u\_{2\text{x}}^2 + u\_2^2 u\_{2\text{x}} \right) &= 0, \end{aligned} \tag{44}$$

$$\begin{aligned} N\_1(u\_1, u\_2) &= a \left( u\_1^2 + u\_2^2 \right)\_{\text{xx}}^{\text{\tiny\text{\tiny}}} u\_{2\text{xx}} + u\_2^3 + u\_1^2 u\_2 + \beta\_3 u\_{1\text{xxx}} + \sigma \left( u\_{1\text{x}}^2 u\_1 + u\_{2\text{x}}^2 u\_1 \right) \\ &+ \theta \left( u\_1^2 u\_{1\text{x}}^2 + u\_2^2 u\_{1\text{x}} \right) = \mathbf{0}, \end{aligned} \tag{45}$$

By applying an inverse operator *L*�1 *t* ¼ Ð*t* 0 ð Þ� *dt* to Eqs. (42) and (43), we get

$$u\_1(\mathbf{x}, t) = u\_1(\mathbf{x}, 0) - L\_t^{-1} N\_2(u\_1(\mathbf{x}, t), u\_2(\mathbf{x}, t)) \tag{46}$$

$$u\_2(\mathbf{x}, t) = u\_2(\mathbf{x}, \mathbf{0}) + L\_t^{-1} N\_1(u\_1(\mathbf{x}, t), u\_2(\mathbf{x}, t)), \tag{47}$$

where *u*1ð Þ¼ *x*, 0 Re ð Þ *u x*ð Þ , 0 and *u*2ð Þ¼ *z*, 0 Imð Þ *q z*ð Þ , 0 .

$$u\_{1,0}(\mathbf{x},t) = u\_1(\mathbf{x},0)$$

$$u\_{2,0}(\mathbf{x},t) = u\_2(\mathbf{x},0) \tag{48}$$

$$u\_{1,k+1}(\mathbf{x},t) = -L\_t^{-1}(N\_{2,k})$$

$$u\_{2,k+1}(\mathbf{x},t) = L\_t^{-1}(N\_{1,k}),$$

$$A\_n = \frac{1}{n!} \frac{d^n}{d\eta^n} N\_j \left(\sum\_{n=0}^\infty \eta^n u\_{1,n}(\mathbf{x},t), \sum\_{n=0}^\infty \eta^n u\_{2,n}(\mathbf{x},t)\right). \tag{49}$$

### **4. Numerical simulations**

### **4.1 Bright solitons**

To depict the ability, reliability and the accuracy of the ADM for Sasa-Satsuma equation for bright solitons where, *a* = 1 10, *b* = 4 100, *β*3 = 1 100, *σ* = 1 10, *θ* = � 173 200, *θ*0 = 0, *ω* = 311 800, and *κ* = 1 2 . The results and the profile of bright soliton shown in Table below. **Figures 4**–**14**, present the plots of exact and approximate solution with their error plots respectively by varying the values of *t* and *x*.

*A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition…   DOI: http://dx.doi.org/10.5772/intechopen.101817*

### **4.2 Dark solitons**

In order to depict ADM, we acknowledge the Sasa-Satsuma equation in investigated in detail where, *a* = 1 10, *b* = 1 2 , *β*3 = 1 5 , *σ* = 1 10, *θ* = � 173 200, *θ*0=0, *ω* = 311 800, and *κ* = 1 2 . The results and the profile of dark soliton shown in Table below. **Figures 15**–**22**, present the plots of exact and approximate solution with their error plots respectively by varying the values of *t* and *x*.

### **4.3** *W* **shaped solitons**

In order to depict ADM, we acknowledge the Sasa-Satsuma equation in investigated in detail where, *a* = 4 100, *b* = 1 2 , *β*3 = 4 100, *σ* = 1 10, *ρ* = ffiffi 3 p 10, *θ* = � 173 200, *θ*0 = 0, *ω* = 311 800,

### **Figure 15.**

*The graph of analytical and numerical solution with absolute error at t* ¼ 0 *and a =* 1 10*,b=* 1 2*, β*3 *=* 1 5*, σ =* 1 10*, θ =* � 173 200*, θ*0 *=* 0*, ω =* 311 800*, κ =* 1 2*.*

### **Figure 16.**

*The graph of analytical and numerical solution with absolute error at t* ¼ 0*:*1 *and a =* 1 *,b=* 1 *, β*3 *=* 1 *, σ =* 1 *, θ =* � 173 *, θ*0 *=* 0*, ω =* 311 *, κ =* 1 *.*

### **Figure 17.**

*The graph of analytical and numerical solution with absolute error at t* ¼ 0*:*2 *and a =* 1 *,b=* 1 *, β*3 *=* 1 *, σ =* 1 *, θ =* � 173 *, θ*0 *=* 0*, ω =* 311 *, κ =* 1 *.*

### **Figure 18.**

*The graph of analytical and numerical solution with absolute error at t* ¼ 0*:*3 *and a =* 1 *,b=* 1 *, β*3 *=* 1 *, σ =* 1 *, θ =* � 173 *, θ*0 *=* 0*, ω =* 311 *, κ =* 1 *.*

*A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition… DOI: http://dx.doi.org/10.5772/intechopen.101817*

### **Figure 19.**

*The graph of analytical and numerical solution with absolute error at x* ¼ 0*:*1 *and a =* 1 *,b=* 1 *, β*3 *=* 1 *, σ =* 1 *, θ =* � 173 *, θ*0 *=* 0*, ω =* 311 *, κ =* 1 *.*

### **Figure 20.**

*The graph of analytical and numerical solution with absolute error at x* ¼ 0*:*2 *and a =* 1 *,b=* 1 *, β*3 *=* 1 *, σ =* 1 *, θ =* � 173 *, θ*0 *=* 0*, ω =* 311 *, κ =* 1 *.*

### **Figure 21.**

*The graph of analytical and numerical solution with absolute error at x* ¼ 0*:*3 *and a =* 1 *,b=* 1 *, β*3 *=* 1 *, σ =* 1 *, θ =* � 173 *, θ*0 *=* 0*, ω =* 311 *, κ =* 1 *.*

### **Figure 22.**

*The graph of analytical and numerical solution with absolute error at t* ¼ 1 *and a =* 1 10*,b=* 1 2*, β*3 *=* 1 5*, σ =* 1 10*, θ =* � 173 200*, θ*0 *=* 0*, ω =* 311 800*, κ =* 1 2*.*

### **Figure 23.**

*The graph of analytical and numerical solution with absolute error at t* ¼ 2 *and a =* 1 10*,b=* 1 2*, β*3 *=* 1 5*, σ =* 1 10*, θ =* � 173 200*, θ*0 *=* 0*, ω =* 311 800*, κ =* 1 2*.*

and *κ* = 1 2 . The results and the profile of W-shaped soliton shown in Table below. **Figures 23**–**34**, present the plots of exact and approximate solution with their error plots respectively by varying the values of *t* and *x*.

*A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition…   DOI: http://dx.doi.org/10.5772/intechopen.101817*

### **5. Conclusion**

In this chapter, SSE has been discussed. First, the undetermined coefficient method has been used. This method secured bright, dark, singular, and *W*-shaped solitons solutions. The method has given a spectrum of solitons. After that, the Adomian decomposition method has been used for the numerical simulations. This is a very powerful method that has given rapid convergence. Along with, error plots have also been given to witness the accuracy of the exact solution. The graphs have also shown the comparison of exact and absolute solution.

### **Conflict of interest**

"The authors declare no conflict of interest".

### **Author details**

Mir Asma Faculty of Science, Institute of Mathematical Science, University of Malaya, Malaysia

\*Address all correspondence to<EMAIL_ADDRESS>
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Hasegawa A, Tappert F. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Applied Physics Letters. 1973;**73**(3):142-144. DOI: 10.1063/1.1654836 [Accessed: November 14, 2016]

[2] Mollenauer LF, Stolen RH, Gordon JP. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Physical Review Letters. 1980;**45**(13):1095-1098. DOI: 10.1103/PhysRevLett.45.1095 [Accessed: November 14, 2016]

[3] Sasa N, Satsuma J. New-type of soliton solutions for a higher-order nonlinear Schrödinger equation. Journal of the Physical Society of Japan. 1991; **60**(2):409-417. DOI: 10.1143/JPSJ.60.409 [Accessed: November 14, 2016]

[4] Akhmediev N, Soto-Crespo JM, Devine N, Hoffmann NP. Rogue wave spectra of the Sasa-Satsuma equation. Physica D: Nonlinear Phenomena. 2015; **294**:37-42. DOI: 10.1016/j. physd.2014.11.006 [Accessed: November 14, 2016]

[5] Adem AR, Ntsime BP, Biswas A, Asma M, Ekici M, Moshokoa SP, et al. Stationary optical solitons with Sasa-Satsuma equation having nonlinear chromatic dispersion. Physics Letters A. 2020;**384**(27):126721. DOI: 10.1016/j. physleta.2020.126721 [Accessed: November 14, 2016]

[6] Wright OC III. Sasa-Satsuma equation unstable plane waves and heteroclinic connections. Chaos, Solitons & Fractals. 2007;**33**: 374-387. DOI: 10.1016/j. chaos.2006.09.034 [Accessed: November 14, 2016]

[7] Asma M, Othman WAM, Wong BR, Biswas A. Optical soliton perturbation with quadratic-cubic nonlinearity by

semi-inverse variational principle. Proceedings of the Romanian Academy Series A. 2017;**18**(4):331-336 [Accessed: November 14, 2016]

[8] Asma M, Othman WAM, Wong BR, Biswas A. Optical soliton perturbation with quadratic-cubic nonlinearity by traveling wave hypothesis. Optoelectronics and Advanced Materials—Rapid Communications. 2017;**11**(9):517-519 [Accessed: November 14, 2016]

[9] Biswas A, Ullah MZ, Asma M, Zhou Q, Moshokoa SP, Belic M. Optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle. Optik. 2017;**139**:16-19. DOI: 10.1016/j.ijleo.2017.03.111 [Accessed: November 14, 2016]

[10] Asma M, Othman WAM, Wong BR, Biswas A. Optical soliton perturbation with quadratic-cubic nonlinearity by the method of undetermined coefficients. Optoelectronics and Advanced Materials. 2017;**19**(11):699-703 [Accessed: November 14, 2016]

[11] Wazwaz AM. Partial Differential Equations and Solitary Waves Theory. 2nd ed. Springer; 2009. ISBN 978-3- 642-00251-9. Available from: https:// link.springer.com/book/10.1007/978-3- 642-00251-9

### **Chapter 4**

## Resonant Optical Solitons in (3 + 1)-Dimensions Dominated by Kerr Law and Parabolic Law Nonlinearities

*Khalil S. Al-Ghafri*

### **Abstract**

This study investigates the optical solitons of of (3+1)-dimensional resonant nonlinear Schrödinger (3D-RNLS) equation with the two laws of nonlinearity. The two forms of nonlinearity are represented by Kerr law and parabolic law. Based on complex transformation, the traveling wave reduction of the governing model is derived. The projective Riccati equations technique is applied to obtain the exact solutions of 3D-RNLS equation. Various types of waves that represent different structures of optical solitons are extracted. These structures include bright, dark, singular, dark-singular and combined singular solitons. Additionally, the obliquity effect on resonant solitons is illustrated graphically and is found to cause dramatic variations in soliton behaviors.

**Keywords:** Optical solitons, 3D-RNLS equation, Kerr law and parabolic law nonlinearities, Projective Riccati equations method, Obliquity influence

### **1. Introduction**

Soliton is one of the important nonlinear waves that has been under intensive investigation in the physical and natural sciences. It has been noticed that solitons play a significant role on describing the physical phenomena in various branches of science, such as optical fibers, plasma physics, nonlinear optics, and many other fields [1–5]. For example, solitons in the field of nonlinear optics are known as optical solitons and have the capacity to transport information through optical fibers over transcontinental and transoceanic distances in a matter of a few femtoseconds [6, 7]. Moreover, it is found that the efficient physical properties of solitons may support the improvement on photonic and optoelectronic devices [8, 9]. Further to this, optical solitons can be exploited widely in optical communication and optical signal processing systems [10, 11].

The formation of solitons is essentially caused due to a delicate balance between dispersion and nonlinearity in the medium. Understanding the dynamics of solitons can be performed through focusing deeply on one model of the nonlinear Schrödinger family of equations with higher order nonlinear terms [12, 13]. Thus, various studies in literatures scrutinized the resonant nonlinear Schrödinger equation which is mainly the governing model that describes soliton propagation and

Madelung fluids in many nonlinear media. Several integration schemes have been implemented to examine the behavior of solitons such as ansatz method, semiinverse variational principle, simplest equation approach, first integral method, functional variable method, sine-cosine function method, (*G*0 *=G*)-expansion method, trial solution approach, generalized extended tanh method, modified simple equation method, and improved extended tanh-equation method. For more details, readers are referred to references [14–25].

The present study concentrates on the investigation of resonant optical solitons in (3 + 1)-dimensions with two types of nonlinear influences, namely, Kerr law and parabolic law nonlinearities. In particular, we shed light on the model of (3 + 1) dimensional resonant nonlinear Schrödinger (3D-RNLS) equation given in the form

$$i\mathbf{Q}\_t + \eta \nabla^2 \mathbf{Q} + \sigma F(s)|\mathbf{Q}|^2 \mathbf{Q} + s\delta \left(\frac{\nabla^2 |\mathbf{Q}|}{|\mathbf{Q}|}\right) \mathbf{Q} = \mathbf{0}, \quad i = \sqrt{-1}, \quad \nabla^2 = \frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial \mathbf{y}^2} + \frac{\partial^2}{\partial \mathbf{z}^2}, \tag{1}$$

where the dependent variable *Q x*ð Þ , *y*, *z*, *t* is a complex-valued wave profile and the independent variables *x*, *y* and *z* stand for spatial coordinates while *t* indicates temporal coordinate. The non-zero constants *η*, *σ*, and *δ* account for the coefficients of the group velocity, non-Kerr nonlinearity, and resonant nonlinearity, respectively. The parameter *s* plays an important role on manipulating the physical properties of distinct media and consequently affecting the behaviors of constructed solitons, see [26].

Here, we will consider two specific cases for the function *F s*ð Þ that represent the effect of nonlinearity in the media. These two nonlinear influences include the Kerr law and parabolic law nonlinearities. Hence, Eq. (1) with the two laws of nonlinearity has the following forms

$$iQ\_t + \eta \nabla^2 Q + \sigma |Q|^2 Q + \delta \left(\frac{\nabla^2 |Q|}{|Q|}\right) Q = 0,\tag{2}$$

and

$$i\mathbf{Q}\_t + \eta \nabla^2 \mathbf{Q} + \left(\sigma |\mathbf{Q}|^2 + \rho |\mathbf{Q}|^4\right) \mathbf{Q} + \delta \left(\frac{\nabla^2 |\mathbf{Q}|}{|\mathbf{Q}|}\right) \mathbf{Q} = \mathbf{0}.\tag{3}$$

The first model given in Eq. (2) is the 3D-RNLS equation dominated by the Kerr law nonlinearity and is found to have applications in the optical fiber and water waves when the refractive index of the light is proportional to the intensity. The second model presented in Eq. (3) is the 3D-RNLS equation with the parabolic law nonlinearity which arises in the context of nonlinear fiber optics.

In literatures, there are some studies that dealt with the 3D-RNLS equation to find exact solutions. For example, Ferdous et al. [27] investigated the conformable time fractional 3D-RNLS equation with Kerr and parabolic law nonlinearities. Different structures of oblique resonant optical solitons have been obtained by using the generalized exp ð Þ �Φð Þ*ξ* -expansion method. Furthermore, Sedeeg et al. [28] studied the two models of 3D-RNLS equation given in (2) and (3) by applying the modified extended tanh method. Optical soliton solutions including dark, singular and combo solitons are extracted in addition to periodic solutions. Moreover, the exact solutions of the 3D-RNLS equation with Kerr law nonlinearity given in (2) has been examined by Hosseini et al. [29] by exploiting the new expansion methods based on the Jacobi elliptic equation. Recently, Hosseini et al. [30] studied the optical solitons and modulation instability of the models given in (2) and (3).

*Resonant Optical Solitons in (3 + 1)-Dimensions Dominated by Kerr Law… DOI: http://dx.doi.org/10.5772/intechopen.100469*

Various forms of optical solitons are derived with the aid of the exp *a* and hyperbolic function techniques.

The aim of current work is to derive the optical solitons of 3D-RNLS equation presented in (2) and (3). The mathematical technique applied to solve the models is based on a finite series expressed in terms of the solution of projective Riccati equations. The paper is organized as follows. In Section 2, we analyze the idea of implementing the proposed method. In Section 3, the traveling wave reduction of (2) and (3) is extracted. Then, Section 4 displays the derivation of resonant optical solitons in (3 + 1)-dimensions. In Section 5, the main outlook of results and remarks are presented. Finally, the conclusion of work is given in Section 6.

### **2. Elucidation of solution method**

Consider a nonlinear partial differential equation (NLPDE) for *Q x*ð Þ , *y*, *z*, *t* in the form

$$P\left(Q, Q\_{\nu}, Q\_{\nu}, Q\_{\gamma}, Q\_{\omega}, Q\_{\infty}, Q\_{\eta\gamma}, Q\_{\pm\pi}, \dots\right) = 0,\tag{4}$$

where *P* is a polynomial in its arguments.

Since we seek for exact traveling wave solutions, we introduce the wave variables

$$Q(\mathbf{x}, t) = q(\xi), \qquad \xi = \mathbf{x}\cos a + \mathbf{y}\cos \beta + \mathbf{z}\cos \gamma + ct. \tag{5}$$

Inserting (5) into Eq. (4), one can find the following ordinary differential equation (ODE)

$$H\left(q, q', q'^{\prime}, q^{\prime\prime}, \dots\right) = \mathbf{0},\tag{6}$$

where prime denotes the derivative with respect to *ξ*. Then, integrate Eq. (6), if possible, to reduce the order of differentiation.

Now, assume that the solution of Eq. (6) can be expressed in the finite series of the form

$$U(\xi) = a\_0 + \sum\_{l=1}^{m} \left( a\_l f^l(\xi) + b g^l(\xi) \right), \tag{7}$$

where *a*0, *al*, *bl*,ð Þ *l* ¼ 1, 2, … , *m* are constants to be identified. The parameter *m*, which is a positive integer, can be determined by balancing the highest order derivative term with the highest order nonlinear term in Eq. (6).

The variables *f*ð Þ*ξ* , *g*ð Þ*ξ* satisfy the equations

$$f'(\xi) = eA g^2(\xi), \qquad g'(\xi) = -A f(\xi) g(\xi) - \frac{B}{A} g(\xi) (R - B f(\xi)),$$

$$g^2(\xi) = e \left[ \frac{1}{A^2} (R - B f(\xi))^2 - f^2(\xi) \right],$$

where *A* and *B* are arbitrary constants and *ε* ¼ �1. The third equation in the system (8) represents the first integral which gives the relation between the functions *f*ð Þ*ξ* and *g*ð Þ*ξ* .

The set of Eqs. (8) is found to admit the following solutions

$$\,\_1f\_1(\xi) = \frac{R\tanh\left(R\xi\right)}{A + B\tanh\left(R\xi\right)}, \qquad \,\_1g\_1(\xi) = \frac{R\text{sech}\left(R\xi\right)}{A + B\tanh\left(R\xi\right)},\tag{9}$$

demands *ε* ¼ 1.

$$f\_{\mathfrak{L}}(\xi) = \frac{R \coth(R\xi)}{A + B \coth(R\xi)}, \qquad g\_{\mathfrak{L}}(\xi) = \frac{R \csc(R\xi)}{A + B \coth(R\xi)},\tag{10}$$

implies *ε* ¼ �1.

$$f\_{\mathfrak{J}}(\xi) = \frac{A}{AC + \left(A^2 - B^2\right)\xi}, \qquad g\_{\mathfrak{J}}(\xi) = \frac{\sqrt{-\varepsilon\left(A^2 - B^2\right)}}{AC + \left(A^2 - B^2\right)\xi}, \tag{11}$$

provided *R* ¼ 0, where *C* is an arbitrary constant.

The substitution of (7) along with (8) into Eq. (6) generates a polynomial in *f i* ð Þ*ξ g j* ð Þ*ξ* . Equating each coefficient of *f i* ð Þ*ξ g j* ð Þ*ξ* in this polynomial to zero, yields a set of algebraic equations for *ai*, *b j*. Solving this system of equations, we can obtain many exact solutions of Eq. (4) according to (9)–(11).

### **3. Traveling wave reduction for Eqs. (2) and (3)**

In order to tackle the complex models of 3D-RNLS equation with Kerr law and parabolic law nonlinearities given in (2) and (3), we embark on analyzing their structures by using the wave transformation of the form

$$Q(\mathbf{x},t) = q(\xi)e^{i\boldsymbol{\wp}},\tag{12}$$

where

$$\xi = \mathbf{x}\cos a + y\cos \beta + \mathbf{z}\cos \gamma + \nu t,\ \boldsymbol{\varrho} = \mathbf{x}(\mathbf{x}\cos a + y\cos \beta + \mathbf{z}\cos \gamma) + a\mathbf{t}.\tag{13}$$

### **3.1 Traveling wave reduction for Eq. (2)**

Applying transformation (12), the 3D-RNLS equation with Kerr law nonlinearity given in (2) is broken down into real and imaginary parts as

$$\begin{cases} \left(\cos^2 a + \cos^2 \beta + \cos^2 \gamma \right) (\eta + \delta) q^{\eta \prime} - \left(\omega + \eta \kappa^2 (\cos^2 a + \cos^2 \beta + \cos^2 \gamma) \right) q + \sigma q^3 \\ = 0, \end{cases} \tag{14}$$

and

$$(\nu + 2\eta\kappa(\cos^2 a + \cos^2 \beta + \cos^2 \gamma))q' = 0. \tag{15}$$

From Eq. (15), we obtain

$$
\omega = -2\lambda\eta\kappa,\tag{16}
$$

*Resonant Optical Solitons in (3 + 1)-Dimensions Dominated by Kerr Law… DOI: http://dx.doi.org/10.5772/intechopen.100469*

where *λ* ¼ cos 2*α* þ cos 2*β* þ cos 2*γ*. Hence, Eq. (14) reduces to the form

$$
\lambda(\eta+\delta)q^{\prime\prime} - (\alpha+\lambda\eta\kappa^2)q + \sigma q^3 = \mathbf{0}.\tag{17}
$$

### **3.2 Traveling wave reduction for Eq. (3)**

Similarly, we utilize the wave transformation (12) to the 3D-RNLS equation with parabolic law nonlinearity given in (3) which is decomposed to real and imaginary parts as

$$
\lambda(\eta+\delta)q^{\prime\prime} - (\alpha+\lambda\eta\kappa^2)q + \sigma q^3 + \rho q^5 = 0,\tag{18}
$$

and

$$(\nu + 2\lambda\eta\kappa)q' = 0.\tag{19}$$

From Eq. (19), we come by the expression given in (16). To seek a closed form solution, the structure of Eq. (18) has to be rearranged. Thus, we multiply Eq. (18) by *q*0 and integrate with respect to *ξ* to arrive at

$$
\lambda(\eta+\delta)q'^2 - \left(\alpha+\lambda\eta\kappa^2\right)q^2 + \frac{\sigma}{2}q^4 + \frac{\rho}{3}q^6 + 2\mu = 0,\tag{20}
$$

where *μ* is the integration constant. For convenience, we make use of the variable transformation given as

$$q^2 = V,\tag{21}$$

which leads to *q*02 ¼ *V*02 *=*4*V*. Thus, Eq. (20), after manipulating, becomes

$$
\lambda(\eta+\delta)V^{\eta^2} + 8\mu V - 4\left(\omega + \lambda\eta\kappa^2\right)V^2 + 2\sigma V^3 + \frac{4}{3}\rho V^4 = 0. \tag{22}
$$

### **4. Optical soliton solutions of 3D-RNLS equation with Kerr law and parabolic law nonlinearities**

Now, we aim to employ the projective Riccati equations method given in Section 2 to extract the exact resonant optical soliton solutions with Kerr law and parabolic law nonlinearities for 3D-RNLS equations given in (2) and (3). Basically, the proposed technique will be implemented to Eqs. (17) and (20) and then their obtained solutions will be inserted into (12) so as to derive the optical solitons of the models discussed in this work.

### **4.1 Oblique resonant solitons of 3D-RNLS equation with Kerr law nonlinearity**

According to the expansion given in (7) and the balance between the terms *q*00 and *q*3, the solution of Eq. (17) reads

$$q(\xi) = a\mathbf{o} + a\mathbf{y}f(\xi) + b\mathbf{y}g(\xi). \tag{23}$$

Substituting (23) together with Eqs. (8) into Eq. (17) gives rise to an equation having different powers of *f i g j* . Collecting all the terms with the same power of *f i g j* together and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations simultaneously leads to the following results.

**Set I.** If *ε* ¼ 1, then the following cases of solutions are retrieved.

$$\textbf{Case I1.}\ \boldsymbol{a}\_0 = \boldsymbol{a}\_1 = \boldsymbol{0}, \boldsymbol{b}\_1 = \pm \sqrt{\frac{2\boldsymbol{\lambda}(\eta + \delta)\left(\boldsymbol{A}^2 - \boldsymbol{B}^2\right)}{\sigma}}, \boldsymbol{o} = \boldsymbol{\lambda}\left( [\eta + \delta]\boldsymbol{R}^2 - \eta \kappa^2 \right).$$

$$Q(\mathbf{x}, y, \mathbf{z}, t) = \pm R \sqrt{\frac{2l(\eta + \delta)(\mathbf{A}^2 - \mathbf{B}^2)}{\sigma}} \frac{\mathrm{sech}(R[\mathbf{x}\cos a + y\cos \beta + \mathbf{z}\cos \gamma - 2l\eta \mathbf{x}t])}{A + B\tanh\left(R[\mathbf{x}\cos a + y\cos \beta + \mathbf{z}\cos \gamma - 2l\eta \mathbf{x}t]\right)}} \mathrm{e}^{i\varphi},\tag{24}$$

$$\begin{aligned} \text{where } &\lambda\sigma(\eta+\delta)\left(A^2 - B^2\right) > 0 \text{ and } \rho = \kappa(x\cos\alpha + y\cos\beta + z\cos\gamma) + \\ &\lambda\left(|\eta+\delta|R^2 - \eta\kappa^2\right)t. \\ \text{Case I2. } a\_0 = \pm \frac{R\theta}{A}\sqrt{-\frac{2l(\eta+\delta)}{\sigma}}, a\_1 = \pm \frac{A^2 - B^2}{A}\sqrt{-\frac{2l(\eta+\delta)}{\sigma}}, b\_1 = 0, \\ \text{or } &\rho = -\lambda\left(2|\eta+\delta|R^2 + \eta\kappa^2\right). \end{aligned}$$

$$\begin{aligned} Q(x, y, z, t) &= \pm R\sqrt{-\frac{2\lambda(\eta+\delta)}{\sigma}} \frac{B+A\tanh\left(R[x\cos\alpha + y\cos\beta + z\cos\gamma - 2l\eta\kappa t]\right)}{A+B\tanh\left(R[x\cos\alpha + y\cos\beta + z\cos\gamma - 2l\eta\kappa t]\right)}e^{i\rho}, \\ &\tag{25} \end{aligned}$$

where *λσ η*ð Þ þ *δ* < 0 and *φ* ¼ *κ*ð*x* cos *α* þ *y* cos *β* þ *z* cos *γ*Þ � *λ* 2½ � *η* þ *δ R*2 þ *ηκ*2 � �*t*.  **Set II.** If *ε* ¼ �1, then the following cases of solutions are generated. **Case II1.** *a*0 ¼ *a*1 ¼ 0, *b*1 ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2*λ η*ð Þ þ*δ A*2�*B*2 ð Þ *σ* q , *ω* ¼ *λ η*½ � þ *δ R*2 � *ηκ*2 � �.

$$Q(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) = \pm R \sqrt{-\frac{2\lambda(\eta + \delta)\left(A^2 - B^2\right)}{\sigma}} \frac{\mathrm{csch}(R[\mathbf{x}\cos a + y\cos \beta + z\cos \gamma - 2\lambda \eta \mathbf{x}])}{A + B\mathrm{csch}(R[\mathbf{x}\cos a + y\cos \beta + z\cos \gamma - 2\lambda \eta \mathbf{x}])}} e^{i\boldsymbol{\wp}},\tag{27}$$

where *λσ η*ð Þ þ *δ A*2 � *B*2 � �< 0 and *φ* ¼ *κ*ð*x* cos *α* þ *y* cos *β* þ *z* cos *γ*Þ þ *λ η*½ � þ *δ R*2 � *ηκ*2 � �*t*.

**Case II2.** *a*0 ¼ � *BR A* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2*λ η*ð Þ þ*δ σ* q , *a*1 ¼ � *A*2 �*B*2 *A* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2*λ η*ð Þ þ*δ σ* q , *b*1 ¼ 0, *ω* ¼ �*λ* 2½ � *η* þ *δ R*2 þ *ηκ*2 � �.

*Resonant Optical Solitons in (3 + 1)-Dimensions Dominated by Kerr Law… DOI: http://dx.doi.org/10.5772/intechopen.100469*

$$Q(\mathbf{x}, y, \mathbf{z}, t) = \pm R\sqrt{-\frac{2\lambda(\eta + \delta)}{\sigma}} \frac{B + A\coth(R[\mathbf{x}\cos a + y\cos\beta + z\cos\gamma - 2\imath\eta \mathbf{x}t])}{A + B\coth(R[\mathbf{x}\cos a + y\cos\beta + z\cos\gamma - 2\imath\eta \mathbf{x}t])}e^{i\rho},\tag{28}$$

$$\text{where } \lambda\sigma(\eta+\delta) < 0 \text{ and } \rho = \kappa(\ge \cos a + y \cos \beta + z \cos \gamma) - \lambda(2[\eta + \delta]R^2 + \eta \kappa^2)t.$$

$$\textbf{Case II3.}, a\_0 = \pm \frac{\pi R}{A} \sqrt{-\frac{\lambda(\eta+\delta)}{2\sigma}}, a\_1 = \pm \frac{A^2 - B^2}{A} \sqrt{-\frac{\lambda(\eta+\delta)(A^2 - B^2)}{2\sigma}},$$

$$\boldsymbol{\rho} = -\lambda \left( [\eta + \delta] \frac{R^2}{2} + \eta \kappa^2 \right).$$

$$Q(\boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z}, t) = \pm R \sqrt{-\frac{\lambda(\eta + \delta)}{2\sigma}} \begin{cases} B + A \coth(R[\boldsymbol{x}\cos\alpha + \boldsymbol{y}\cos\beta + \boldsymbol{z}\cos\gamma - 2\lambda\eta \boldsymbol{x}t]) \\ A + B \coth(R[\boldsymbol{x}\cos\alpha + \boldsymbol{y}\cos\beta + \boldsymbol{z}\cos\gamma - 2\lambda\eta \boldsymbol{x}t]) \end{cases}$$

$$\pm \frac{\sqrt{A^2 - B^2} \text{csch}(R[\boldsymbol{x}\cos\alpha + \boldsymbol{y}\cos\beta + \boldsymbol{z}\cos\gamma - 2\lambda\eta \boldsymbol{x}t])}{A + B \coth(R[\boldsymbol{x}\cos\alpha + \boldsymbol{y}\cos\beta + \boldsymbol{z}\cos\gamma - 2\lambda\eta \boldsymbol{x}t])} \end{cases} \tag{29}$$

where *λσ η*ð Þ þ *δ* < 0, *A*2 > *B*2 and *φ* ¼ *κ*ð*x* cos *α* þ *y* cos *β* þ *z* cos *γ*Þ � *λ η*½ � þ *δ R*2 2 þ *ηκ*2 � �*t*. **Set III.** If *R* ¼ 0, then the following cases of solutions are created. **Case III1.** *a*0 ¼ *a*1 ¼ 0, *b*1 ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*λ η*ð Þ þ*δ A*2�*B*2 ð Þ *εσ* q , *ω* ¼ �*ληκ*2 .

$$\begin{aligned} \text{Case III2. } a\_0 = a\_1 - \alpha, a\_1 = \pm \sqrt{\frac{\alpha}{A}}, \quad \text{where} \quad \alpha = -\eta\kappa. \\ \text{Case III2. } a\_0 = 0, a\_1 = \pm \frac{A^2 - B^2}{A} \sqrt{-\frac{2\dot{\iota}(\eta + \delta)}{\sigma}}, b\_1 = 0, \alpha = -\lambda\eta\kappa^2. \\ \text{Case III3. } a\_0 = 0, a\_1 = \pm \frac{A^2 - B^2}{A} \sqrt{-\frac{\dot{\iota}(\eta + \delta)}{2\sigma}}, b\_1 = \pm \sqrt{\frac{\dot{\iota}(\eta + \delta)\left(A^2 - B^2\right)}{2\sigma}}, \alpha = -\lambda\eta\kappa^2. \\ \text{Herein, these three cases in the Set III provide the solution of the form} \end{aligned}$$

$$Q(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) = \pm \sqrt{-\frac{2\lambda(\eta + \delta)}{\sigma}} \frac{(A^2 - B^2)}{AC + (A^2 - B^2)(\mathbf{x} \cos a + \mathbf{y} \cos \beta + \mathbf{z} \cos \gamma - 2\lambda \eta \mathbf{x} t)} e^{i\mathbf{y}\cdot\mathbf{z}},\tag{30}$$

where *λσ η*ð Þ þ *δ* < 0 and *φ* ¼ *κ*ð*x* cos *α* þ *y* cos *β* þ *z* cos *γ*Þ � *ληκ*2 *t*.

### **4.2 Oblique resonant solitons of 3D-RNLS equation with parabolic law nonlinearity**

Based on the expansion given in (7), we consider that the solution to Eq. (22) takes the form

$$q(\xi) = a\_0 + \sum\_{l=1}^{2} \left( a\_l f^l(\xi) + b\_l \mathbf{g}^l(\xi) \right). \tag{31}$$

Inserting (31) together with Eqs. (8) into Eq. (22) gives rise to an equation having different powers of *f i g j* . Collecting all the terms with the same power of *f i g j* together and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations simultaneously leads to the following results.

**Set I.** If *ε* ¼ 1, then the following cases of solutions are obtained.

 $\textbf{Case I1.}$   $b\_1 = a\_2 = b\_2 = 0, a\_0 = \frac{-Ra\_1}{(A+B)}, a\_1 = \pm \frac{A^2 - B^2}{2A}\sqrt{-\frac{3\ell(\eta+\delta)}{\rho}}$

*The Nonlinear Schrödinger Equation*

$$o = \frac{\lambda[(\eta + \delta)(A + B)^2 a\_0^2 - \eta^2 a\_1^2]}{a\_1^2}, \sigma = \frac{2\lambda(\eta + \delta)(A - B)(A + B)^2 a\_0}{A a\_1^2}, \mu = 0.$$

$$Q(\mathbf{x}, y, z, t) = \left[ \mp \frac{R}{2} \sqrt{-\frac{3\lambda(\eta + \delta)}{\rho}} \left\{ \mathbf{1} - \frac{B + A \tanh\left(R[\mathbf{x} \cos a + y \cos \theta + x \cos \gamma - 2\lambda \eta \mathbf{x}]\right)}{A + B \tanh\left(R[\mathbf{x} \cos a + y \cos \theta + x \cos \gamma - 2\lambda \eta \mathbf{x}]\right)} \right\} \right]^{\frac{1}{2}} e^{i\rho}, \tag{32}$$

$$\begin{split} & \text{where } \lambda \rho (\eta + \delta) < 0 \text{ and } \rho = \kappa (\propto \cos a + y \cos \beta + z \cos \gamma) + \lambda \left( [\eta + \delta] R^2 - \eta \kappa^2 \right) t. \\ & \mathbf{Case I2.} \, b\_1 = a\_2 = b\_2 = 0, \, a\_0 = \frac{a a\_1}{(A - B)}, \, a\_1 = \pm \frac{A^2 - B^2}{2A} \sqrt{-\frac{3 \lambda (\eta + \delta)}{\rho}}, \\ & \boldsymbol{\sigma} = \frac{\lambda \left( (\eta + \delta)(A - B)^2 a\_0^2 - \eta \kappa^2 a\_1^2 \right)}{a\_1^2}, \, \boldsymbol{\sigma} = \frac{2 \boldsymbol{\lambda} (\eta + \delta)(A + B)(A - B)^2 a\_0}{A a\_1^2}, \, \boldsymbol{\mu} = \mathbf{0}. \end{split}$$

$$Q(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) = \left[ \pm \frac{R}{2} \sqrt{-\frac{3\lambda(\eta + \delta)}{\rho}} \left\{ \mathbf{1} + \frac{B + A \tanh\left(R[\mathbf{x} \cos a + y \cos \beta + x \cos y - 2\lambda \eta \mathbf{z}]\right)}{A + B \tanh\left(R[\mathbf{x} \cos a + y \cos \beta + x \cos y - 2\lambda \eta \mathbf{z}]\right)} \right\} \right]^{\frac{1}{2}},\tag{33}$$

$$\begin{aligned} \text{where } &\lambda\rho(\eta+\delta)<0 \text{ and } \rho=\kappa(\mathbf{x}\cos a+\boldsymbol{y}\cos\beta+\boldsymbol{z}\cos\gamma)+\lambda\{ |\eta+\delta|R^{2}-\eta\kappa^{2} \}t. \\\\ \textbf{Case I3.} \, a\_{1}=a\_{2}=b\_{2}=0, \, a\_{0}=\pm\frac{\beta b\_{1}}{\sqrt{A^{2}-B^{2}}}, \, b\_{1}=\pm\frac{1}{2}\sqrt{\frac{3\lambda(\eta+\delta)\left(A^{2}-B^{2}\right)}{\rho}}, \\\ \boldsymbol{a}=-\frac{\boldsymbol{\lambda}\left(\boldsymbol{\xi}(\eta+\delta)\left(A^{2}-B^{2}\right)a\_{0}^{2}+4\eta a^{2}b\_{1}^{2}\right)}{4b\_{1}^{2}}, \,\sigma=-\frac{2\boldsymbol{\lambda}(\eta+\delta)\left(A^{2}-B^{2}\right)a\_{0}}{b\_{1}^{2}}, \,\mu=-\frac{\boldsymbol{\lambda}(\eta+\delta)\left(A^{2}-B^{2}\right)a\_{0}^{3}}{4b\_{1}^{2}}. \end{aligned} \tag{3.27}$$

$$\boldsymbol{Q}(\boldsymbol{x},\boldsymbol{y},\boldsymbol{z},t)=\left[\pm\frac{\boldsymbol{R}}{2}\sqrt{\frac{3\lambda(\eta+\delta)}{\rho}}\bigg{[}1\pm\frac{\sqrt{A^{2}-B^{2}}\mathrm{sech}\left(R[\boldsymbol{x}\cos\boldsymbol{a}+\boldsymbol{y}\cos\boldsymbol{\beta}+\boldsymbol{z}\cos\boldsymbol{y}-2\boldsymbol{\lambda}\eta\boldsymbol{x}]\right)]}\right]\right]^{\frac{1}{2}}\boldsymbol{e}^{i\phi}, \,\mu>-\frac{\left(\frac{\boldsymbol{R}}{2}\sqrt{\frac{3\lambda(\eta+\delta)}{\rho}}\right)^{2}\left(\frac{\$$

where *λρ η*ð Þ þ *δ* >0, *A*2 >*B*2 and *φ* ¼ *κ*ð*x* cos *α* þ *y* cos *β* þ *z* cos *γ*Þ � *λ η*½ � þ *δ* 5*R*2 4 þ *ηκ*2 � �*t*. **Case I4.** *a*2 ¼ *b*2 ¼ 0, *a*0 ¼ � *Rb*1 *A A*ð Þ þ*B* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *A*2 � *B*2 p , *a*1 ¼ � *b*1 *A* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *A*2 � *B*2 p , *b*1 ¼ � 1 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3*λ η*ð Þ þ*δ A*2 �*B*2 ð Þ *ρ* r , *ω* ¼ � *λ η*ð Þ þ*δ* ð Þ *A*þ*B A*2 *a*2 0þ4*ηκ*2ð Þ *A*�*B b*2 ð Þ1 4ð Þ *A*�*B b*2 1 , *σ* ¼ � *λ η*ð Þ þ*δ* ð Þ *A*þ*B Aa*0 2*b*2 1 , *μ* ¼ 0.  DOI: http://dx.doi.org/10.5772/intechopen.100469*

$$Q(\mathbf{x}, y, z, t) = \left[ \pm \frac{R}{4A} \sqrt{\frac{3\lambda(\eta + \delta)(A^2 - B^2)}{\rho}} \right] \left\{ \frac{\sqrt{-\left(A^2 - B^2\right)} \tanh\left(R[\mathbf{x}\cos a + y\cos\theta + z\cos\chi - 2i\eta xt]\right)}{A + B\tanh\left(R[\mathbf{x}\cos a + y\cos\theta + z\cos\chi - 2i\eta xt]\right)} \right. \tag{14.59}$$

$$\pm \frac{A\operatorname{sech}\left(R[\mathbf{x}\cos a + y\cos\theta + z\cos\chi - 2i\eta xt]\right)}{A + B\tanh\left(R[\mathbf{x}\cos a + y\cos\theta + z\cos\chi - 2i\eta xt]\right)} + \frac{\sqrt{-\left(A^2 - B^2\right)}}{(A - B)} \right\} \int\_{-\rho}^{\frac{\rho}{\sqrt{-\left(A^2 - B^2\right)}}} \tag{14.50}$$

$$\begin{split} \text{where } &\lambda\rho(\eta+\delta) < 0, A^2 < B^2 \text{ and } \rho = \kappa(\ge \cos a + y\cos \beta + z\cos \gamma) + \\ &\lambda\left([\eta+\delta]\frac{R^2}{4} - \eta\kappa^2\right)t. \\ \mathbf{Case I6, } a\_0 = -\frac{\left(2i(\eta+\delta)A^2 + \sigma b\_2\right)R^2}{\sigma A^2}, a\_1 = \frac{2RBb\_2}{A^2}, b\_1 = \pm \frac{A^2a\_0 + R^2b\_2}{RA^2}\sqrt{A^2 - B^2}, a\_2 = \frac{\left(\sqrt{A^2 - B^2}\right)^2}{A^2}, \\ \frac{\left(A^2 - B^2\right)b\_2}{A^2}, a = -\lambda\left([\eta+\delta]\frac{5R^2}{4} + \eta\kappa^2\right), \rho = \frac{3\sigma^2}{16d(\eta+\delta)R^2}, \mu = \frac{\lambda^2(\eta+\delta)^2R^4}{2\sigma}. \end{split}$$

$$Q(\mathbf{x}, \mathbf{y}, \mathbf{z}, t) = \left[ -\frac{2\lambda(\eta + \delta)R^2}{\sigma} \left\{ 1 \pm \frac{\sqrt{A^2 - B^2} \text{sech}(R[\mathbf{x}\cos\alpha + \mathbf{y}\cos\beta + \mathbf{z}\cos\gamma - 2\lambda\eta \mathbf{x}t])}{A + B\tanh\left(R[\mathbf{x}\cos\alpha + \mathbf{y}\cos\beta + \mathbf{z}\cos\gamma - 2\lambda\eta \mathbf{x}t]\right)} \right\} \right]^{\frac{1}{2}} e^{i\boldsymbol{\wp}},\tag{37}$$

$$\begin{split} & \text{where } A^2 > B^2 \text{ and } \rho = \kappa (\mathbf{x} \cos \alpha + \mathbf{y} \cos \beta + \mathbf{z} \cos \gamma) - \lambda \left( [\eta + \delta] \frac{\mathfrak{A} \mathbb{R}^2}{4} + \eta \mathbb{R}^2 \right) \text{t.} \\ & \mathbf{C} \mathbf{a} \mathbf{e} \mathbf{I} \mathbf{7}, b\_1 = \mathbf{0}, a\_0 = \frac{\left( A^2 a\_1 - (A + B) \mathbb{R} b\_1 \right) \mathbb{R}}{(A - B) \mathbf{A}^2}, a\_1 = \frac{2 \left( \mathbb{A} (\eta + \delta) \left( A^2 - \mathbb{B}^2 \right) A + \sigma \mathrm{B} b\_1 \right) \mathbb{R}}{\sigma \mathbf{A}^2}, a\_2 = \frac{\left( A^2 - \mathbb{B}^2 \right) b\_1}{\mathbf{A}^2}, \\ & \alpha = \lambda \left( [\eta + \delta] R^2 - \eta \kappa^2 \right), \rho = \frac{3 \lambda (\eta + \delta) \left( A^2 - \mathbb{B}^2 \right)^2 A^2}{4 \left( A^2 a\_1 - 2 \mathrm{B} \mathrm{B} b\_1 \right)^2}, \mu = \mathbf{0}. \end{split}$$

$$Q(\mathbf{x}, y, z, t) = \left[ \frac{2\lambda(\eta + \delta)R^2}{\sigma} \left\{ 1 + \frac{B + A \tanh\left(R[\mathbf{x} \cos a + y \cos \beta + z \cos \gamma - 2l\eta \mathbf{x}t]\right)}{A + B \tanh\left(R[\mathbf{x} \cos a + y \cos \beta + z \cos \gamma - 2l\eta \mathbf{x}t]\right)} \right\} \right]^{\frac{1}{2}} e^{i\eta},\tag{38}$$

$$\begin{split} \text{where } & \boldsymbol{\rho} = \kappa (\boldsymbol{x} \cos \boldsymbol{a} + \boldsymbol{y} \cos \boldsymbol{\beta} + \boldsymbol{z} \cos \boldsymbol{\gamma}) + \lambda \{ [\boldsymbol{\eta} + \delta] \boldsymbol{R}^{2} - \eta \boldsymbol{\kappa}^{2} \} \boldsymbol{t}, \\ \mathbf{Case \ \mathbf{I} \ } & \mathbf{B}, \ b\_{1} = \mathbf{0}, \ a\_{0} = -\frac{\{ \boldsymbol{A}^{2} \boldsymbol{a}\_{1} + (\boldsymbol{A} - \boldsymbol{B}) \boldsymbol{b} \boldsymbol{b}\_{2} \} \boldsymbol{R}}{\{ \boldsymbol{A} + \boldsymbol{B} \} \boldsymbol{A}^{\dagger}}, \boldsymbol{a}\_{1} = -\frac{2 \{ \boldsymbol{\lambda} (\boldsymbol{\eta} + \delta) \{ \boldsymbol{A}^{2} - \boldsymbol{B}^{2} \} \boldsymbol{A} - \sigma \boldsymbol{B} \boldsymbol{b}\_{2} \} \boldsymbol{R}}{\{ \boldsymbol{A}^{2} \} \boldsymbol{A}^{\dagger}}, \\ & \boldsymbol{a}\_{2} = \frac{\{ \boldsymbol{A}^{2} - \boldsymbol{B}^{2} \} \boldsymbol{b}\_{2}}{\boldsymbol{A}^{\dagger}}, \\ & \boldsymbol{a} = \boldsymbol{\lambda} \{ [\boldsymbol{\eta} + \delta] \boldsymbol{R}^{2} - \eta \boldsymbol{\kappa}^{2} \}, \boldsymbol{\rho} = -\frac{3 \boldsymbol{\lambda} (\boldsymbol{\eta} + \delta) \{ \boldsymbol{A}^{2} - \boldsymbol{B}^{2} \}^{\dagger} \boldsymbol{A}^{2}}{\{ \boldsymbol{A}^{2} \boldsymbol{a}\_{1} - 2 \boldsymbol{B} \boldsymbol{b} \boldsymbol{b}\_{2} \}^{\dagger}}, \boldsymbol{\mu} = \mathbf{0}. \end{split}$$

$$Q(\mathbf{x}, y, \mathbf{z}, t) = \left[ \frac{2\boldsymbol{\lambda}(\eta + \delta)R^2}{\sigma} \left\{ 1 - \frac{B + A \tanh\left(R[\boldsymbol{x}\cos\alpha + \boldsymbol{y}\cos\beta + \boldsymbol{z}\cos\gamma - 2\lambda\eta\kappa\boldsymbol{t}]\right)}{A + B \tanh\left(R[\boldsymbol{x}\cos\alpha + \boldsymbol{y}\cos\beta + \boldsymbol{z}\cos\gamma - 2\lambda\eta\kappa\boldsymbol{t}]\right)} \right\}^{\frac{1}{2}} e^{i\boldsymbol{\eta}\cdot\boldsymbol{y}},\tag{39}$$

$$\begin{split} \text{where } \boldsymbol{\varrho} &= \kappa (\boldsymbol{\kappa} \cos \boldsymbol{a} + \boldsymbol{y} \cos \boldsymbol{\beta} + \boldsymbol{z} \cos \boldsymbol{\gamma}) + \lambda \big( [\eta + \boldsymbol{\delta}] \boldsymbol{R}^{2} - \eta \boldsymbol{\kappa}^{2} \big) \boldsymbol{t}, \\ \mathbf{Case I9. } \boldsymbol{a}\_{0} &= \frac{(\boldsymbol{\lambda} [\eta + \boldsymbol{\delta}] (\boldsymbol{A} + \boldsymbol{B}) \boldsymbol{A} - 2 \boldsymbol{a} b\_{2}) \boldsymbol{R}^{2}}{2 \sigma \boldsymbol{A}^{2}}, \boldsymbol{a}\_{1} = \frac{(\boldsymbol{A} - \boldsymbol{B}) \boldsymbol{A}^{2} \boldsymbol{a}\_{0} + (\boldsymbol{A} + \boldsymbol{B}) \boldsymbol{R}^{2} b\_{2}}{\boldsymbol{A}^{2} \boldsymbol{R}}, \boldsymbol{b}\_{1} = \\ &\pm \frac{\boldsymbol{\lambda} (\eta + \boldsymbol{\delta}) \boldsymbol{R}}{2 \sigma} \sqrt{- \left( \boldsymbol{A}^{2} - \boldsymbol{B}^{2} \right)}, \boldsymbol{a}\_{2} = \frac{\left( \boldsymbol{A}^{2} - \boldsymbol{B}^{2} \right) b\_{2}}{\boldsymbol{A}^{2}}, \boldsymbol{o} = \boldsymbol{\lambda} \Big{(} \boldsymbol{\eta} + \boldsymbol{\delta} \big| \frac{\boldsymbol{B}^{2}}{4} - \boldsymbol{\eta} \boldsymbol{\kappa}^{2} \big), \boldsymbol{\rho} = -\frac{3 \boldsymbol{\rho}^{2}}{4 \boldsymbol{i} (\eta + \boldsymbol{\delta}) \boldsymbol{R}^{2}}, \boldsymbol{\mu} = \boldsymbol{0}. \end{split}$$

$$Q(x,y,z,t) = \left[\frac{\lambda(\eta+\delta)R^2}{2\sigma}\left\{1 + \frac{B+A\tanh\left(R[x\cos a + y\cos\beta + z\cos\gamma - 2\imath\eta\kappa t]\right)}{A+B\tanh\left(R[x\cos a + y\cos\beta + z\cos\gamma - 2\imath\eta\kappa t]\right)}\right\} \right. \tag{4}$$

$$\pm \frac{\sqrt{-\left(A^2 - B^2\right)}\text{sech}\left(R[x\cos a + y\cos\beta + z\cos\gamma - 2\imath\eta\kappa t]\right)}{A+B\tanh\left(R[x\cos a + y\cos\beta + z\cos\gamma - 2\imath\eta\kappa t]\right)}\right\}\Bigg\}\epsilon^{i\wp},\tag{40}$$

$$\begin{split} \text{where } A^2 < B^2 \text{ and } \rho &= \kappa (\mathbf{x} \cos a + y \cos \beta + \mathbf{z} \cos \gamma) + \lambda \left( [\eta + \delta] \frac{B^2}{4} - \eta \kappa^2 \right) \mathbf{t}. \\ \mathbf{Case I10.}, a\_0 &= \frac{(\lambda (\eta + \delta)(A - B)A - 2a b\_1) \mathbb{R}^2}{2 \sigma A^2}, a\_1 = -\frac{(A + B) A^2 a\_0 + (A - B) \mathbb{R}^2 b\_1}{A^2 \mathbb{R}}, b\_1 = \\ \pm \frac{\lambda (\eta + \delta) \mathbb{R}}{2 \sigma} \sqrt{-\left( A^2 - B^2 \right)}, a\_2 &= \frac{(A^2 - B^2) b\_1}{A^2}, a = \lambda \left( [\eta + \delta] \frac{B^2}{4} - \eta \kappa^2 \right), \rho = -\frac{3 \sigma^2}{4 \lambda (\eta + \delta) \mathbb{R}^2}, \mu = \mathbf{0}. \end{split}$$

$$Q(\mathbf{x}, y, z, t) = \left[ \frac{\lambda(\eta + \delta)R^2}{2\sigma} \left\{ 1 - \frac{B + A \tanh\left(R[\mathbf{x}\cos a + y\cos\beta + z\cos\gamma - 2\lambda\eta\varepsilon]\right)}{A + B \tanh\left(R[\mathbf{x}\cos a + y\cos\beta + z\cos\gamma - 2\lambda\eta\varepsilon]\right)} \right\} \right. \tag{4.10}$$

$$\pm \frac{\sqrt{-\left(A^2 - B^2\right)} \mathrm{sech}\left(R[\mathbf{x}\cos a + y\cos\beta + z\cos\gamma - 2\lambda\eta\varepsilon]\right)}{A + B \tanh\left(R[\mathbf{x}\cos a + y\cos\beta + z\cos\gamma - 2\lambda\eta\varepsilon]\right)} \right\} \Bigg/ \tag{4.11}$$