## Abstract

In this chapter, evolution of light beams in a cubic-quintic-septic-nonical medium is investigated. As the model equation, an extended form of the well-known nonlinear Schrödinger (NLS) equation is taken into account. By the use of a special ansatz, exact analytical solutions describing bright/dark and kink solitons are constructed. The existence of the wave solutions is discussed in a parameter regime. Moreover, the stability properties of the obtained solutions are investigated, and by employing Stuart and DiPrima’s stability analysis method, an analytical expression for the modulational stability is found.

### Keywords

- higher-order nonlinear Schrödinger equation
- spatial solitons
- stability analysis method
- modulational instability
- optical fibers

## 1. Introduction

The study of spatial solitons in the field of fiber-optical communication has attracted considerable interest in recent years. In a uniform nonlinear fiber, soliton can propagate over relatively long distance without any considerable attenuation. The formation of optical solitons in optical fibers results from an exact balancing between the diffraction and/or group velocity dispersion (GVD) and the self-phase modulation (SPM). The theorical prediction of a train of soliton pulses from a continuous-wave (CW) light in optical fibers was first suggested by Hasegawa and Tappert [1, 2] and first experimentally demonstrated by Mollenauer et al. [3] in single-mode fibers in the case of negative GVD, in liquid

In addition to fundamental bright and dark solitons, various other forms and shapes of solitary waves can appear in nonlinear media. Kink solitons, for example, are an important class of solitons which may propagate in nonlinear media exhibiting higher-order effects such as third-order dispersion, self-steepening, higher-order nonlinearity, and intrapulse stimulated Raman scattering. In the setting of nonlinear optics, a kink soliton represents a shock front that propagates undistorted inside the dispersive nonlinear medium [10]. This type of solitons has been studied extensively, both analytically and numerically [11, 12, 13]. These spatial soliton solutions can maintain their overall shapes but allow their widths and amplitudes and the pulse center to change according to the management of the system’s parameters, such as the dispersion, nonlinearity, gain, and so on [14].

The cubic nonlinear Schrödinger equation (CNLSE) has been widely used to model the propagation of light pulse in material’s systems involving third-order susceptibility *parabolic law nonlinearity* and existing in nonlinear media such as the p-toluene sulfonate (PTS) crystals. The parabolic law can closely describe the nonlinear interaction between the high-frequency Langmuir waves and the ion acoustic waves by ponderomotive forces [15, 16], in a region of reduced plasma density, and the nonlinear interaction between Langmuir waves and electrons. In addition, CQ was experimentally proposed as an empirical description of special semiconductor (e.g., AlGaAs, CdS, etc.) waveguides and semiconductor-doped glasses, particularly for the

In recent years, many influential works have devoted to construct exact analytical solutions of CQNLSE, such as the pioneering work of Serkin et al. [20]. In particular, Dai et al. [21, 22, 23, 24, 25] obtained exact self-similar solutions (similaritons), their nonlinear tunneling effects of the generalized CQNLSE, and their higher-dimensional forms with spatially inhomogeneous group velocity dispersion, cubic-quintic nonlinearity, and amplification or attenuation.

Since the measurement of third-, fifth-, and seventh-order nonlinearities of silver nanoplatelet colloids using a femtosecond laser [26], an extension of nonlinear Schrödinger equation including the cubic-quintic-septic nonlinearity was used to model the propagation of spatial solitons. In [27], for example, the authors performed numerical calculations based on higher-order nonlinearity parameters including seventh-order susceptibility

Recently, the study of modulational instability (MI) in non-Kerr media has receiving particular attention. MI is a fundamental and ubiquitous process that appears in most nonlinear systems [6, 9, 34, 35, 36, 37]. This instability is referred to as modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train. During this process, small perturbations upon a uniform intensity beam grow exponentially due to the interplay between nonlinearity and dispersion or diffraction. As a result, under specific conditions, a CW light often breaks up into trains of ultrashort solitons like pulses [9]. To date, there has not been any report of MI in the cubic-quintic-septic-nonical-nonlinear Schrödinger equation (CQSNNLSE).

Our study will be focused on the analysis of solitary wave’s solutions of systems described by the higher-order NLSE named CQSNNLSE. We will discuss the model with higher-order nonlinearities and explore the dynamics of bright, dark, and kink soliton solutions. Finally, the linear stability analysis of the MI is formulated, and the analytical expression of the gain of MI is obtained. Moreover, the typical outcomes of the nonlinear development of the MI are reported.

## 2. Model equation

The dynamics of (1 + 1)-dimensional (one spatial and one temporal variables) spatial optical solitons is the well-known nonlinear Schrödinger equation. If we consider the higher-order effects, an extended model is required, and the propagation of optical pulses through the highly nonlinear waveguides can be described by the CQSNNLSE:

where

For example, Eq. (1) with

To obtain the exact analytic optical solitary-wave solutions of Eq. (1), we can employ the following transformation:

Here,

Upon substituting Eq. (2) into Eq. (1) and separating the real and imaginary parts, one obtains

Eq. (4) represents the evolution of an anharmonic oscillator with an effective potential energy

Integrating Eq. (4) yields

where

and

In order to get the exact soliton solutions, we first rewrite Eq. (6) in a simplified form by using transformation:

By substituting Eq. (8) into Eq. (6), we obtain a new auxiliary equation possessing a sixth-degree nonlinear term:

To solve Eq. (9), we will employ three types of localized solutions named bright, dark, and kink solitons. In the following, we solve Eq. (9) by using appropriate ansatz and obtain alternative types of solitary-wave solutions on a CW background and investigate parameter domains in which these optical spatial solitary waves exist.

## 3. Exact solitary-wave solutions

In this section, we find bright-, dark-, and kink-solitary-wave localized solutions of Eq. (9), by using a special ansatz:

### 3.1 Bright solitary-wave solutions

The bright solitary solutions of Eq. (9) have the form:

where

Substituting the ansatz Eq. (10) into Eq. (9), we obtain the unknown parameters

with parametric conditions

Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1) are of the form:

### 3.2 Dark solitary-wave solutions

The dark solitary solutions of Eq. (9) take the form [40]:

Here

By substituting the ansatz Eq. (14) into Eq. (9), we get the unknown parameters

subject to the parametric conditions

The exact dark solitary-wave solutions on a CW background of Eq. (1) are of the form:

### 3.3 Kink solitary-wave solutions

The kink solitary solutions of Eq. (9) are in the following form:

where

Substituting Eq. (18) into Eq. (9), we get

under the parametric conditions

Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1) are of the form:

The previous three exact solitary-wave solutions (13), (17), and (21) exist for the governing nonical-NLS model due to a balance among diffraction (or dispersion) and competing cubic-quintic-septic-nonical nonlinearities. For better insight, we plot in Figure 1 the intensity profile on top of the related first two exact solution solitons named bright and dark, corresponding to the CQNLS models (with

## 4. Modulational instability of the CW background

One of the essential aspects of solitary waves is their stability on propagation, in particular their ability to propagate in a perturbed environment over an appreciable distance [41]. Unlike the conventional pulses of different forms, the solitons are relatively stable, even in an environment subjected to external perturbations.

The previous three exact solitary-wave solutions given by the expressions (13), (17), and (21) are sitting on a CW background, which may be subject to MI. If this phenomenon occurs, then the CW background will be quickly destroyed, which will inevitably cause the destruction of the soliton. It is therefore of paramount importance to verify whether the condition of the existence of the soliton can be compatible with the condition of the stability of the CW background. Since MI properties can be used to understand the different excitation patterns on a CW in nonlinear systems, in this section, we perform the standard linear stability analysis [9, 34] on a generic CW:

in the system modeled by Eq. (1), where

where

where

The gain attains its peak values when the modulated frequency reaches its optimum value, i.e., its optimum modulation frequency (OMF). The OMF corresponding to the gain spectrum (26) is given by

and the peak value given by

In Figure 2, we have shown the variation of OMF, computed from Eq. (27) as a function of the GVD parameter (

We can observe that the OMF increases (respectively decreases) with the increasing

Figure 3 shows the variation of MI gain as a function of the nonic nonlinearity

The MI gain spectrum in Figure 5 is a constitutive of two symmetrical sidebands which stand symmetrically along the line

## 5. Conclusion

In this chapter, we have investigated the higher-order nonlinear Schrödinger equation involving nonlinearity up to the ninth order. We have constructed exact solutions of this equation by means of a special ansatz. We showed the existence of a family of solitonic solutions: bright, dark, and kink solitons. The conditions on the physical parameters for the existence of this propagating envelope have also been reported. These conditions show a subtle balance among the diffraction or dispersion, Kerr nonlinearity, and quintic-septic-nonical non-Kerr nonlinearities, which has a profound implication to control the wave dynamics. Moreover, by employing Stuart and DiPrima’s stability analysis method, an analytical expression for the MI gain has been obtained. The outcomes of the instability development depend on the nonlinearity and dispersion (or diffraction) parameters. Results may find straightforward applications in nonlinear optics, particularly in fiber-optical communication.

## Conflict of interest

The authors declare no conflict of interest.