Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima ’ s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.


Introduction
The wave process is innumerable in nature. Such familiar examples include water waves, plasma waves, and optical waves and are governed by nonlinear partial differential equations. The study of nonlinear evolution equations helps a lot in understanding certain interesting physical properties posed by themselves, in several physical systems. Recently, an important amount of studies has been related to nonlinear systems having multidegrees of freedom: The well-known nonlinear Schrödinger (NLS) equation is a particular example. The idea behind the NLS equation was originated from the work of Erwin Schrödinger, an Autrichian physician, in 1926 [1]. This equation governs weakly nonlinear and dispersive wave packets in one-dimensional (1D) physical systems. It was first derived, in a general setting, by Benney and Newell in 1967 [2]. Also, it was derived in the study of modulational stability of deep-water waves by Zakharov in 1968 [3]. Afterward, Hasegawa and Tappert (1973) showed that the same equation governs light-pulse propagation in optical fibers [4]. For instance, the cubic-NLS has been widely used to model the propagation of light pulse in material's systems involving third-order susceptibility χ 3 ð Þ [5][6][7]. In the same context, the nonlinear interaction between the high-frequency Langmuir waves and the ion-acoustic waves by ponderomotive forces [8,9] in a region of reduced plasma density, and the nonlinear interaction between Langmuir waves and electrons, were described by the "parabolic law nonlinearity" (cubic-quintic CQ) that existing in nonlinear media such as the CdS x Se 1Àx À doped glass [10,11], the poly-toluene sulfonate (PTS) crystals, special semiconductor waveguides (e.g., AlGaAs, CdS) [12]. Furthermore, Serkin et al. [13], Dai et al. [14], and others have devoted pioneering works in order to analyze the dynamical propagation of light pulse in CQ-nonlinear media, by considering the CQ-NLS equation.
Our study will be focused on a nano-optical fiber-system, described by the following extended perturbed NLS equation (integrable equation named as Biswas-Arshed model), involving power-law nonlinearity and higher-order dispersions [20][21][22]: where the complex-valued function ψ z, x ð Þ is designated for waveform, which depends on the temporal variable z and the spatial variable x; a 1 and a 2 are, respectively, the group velocity dispersion (GVD) and spatiotemporal dispersion coefficients (STD). b 1 and b 2 correspond to the coefficient of power-law nonlinearity; α accounts for the inter-modal dispersion. γ and δ account for the selfsteepening perturbation terms, while σ, λ, and θ provide the effect of nonlinear dispersion coefficient. Finally, n denotes the strength of the power-law nonlinearity.
This model is relevant to some applications in which higher-order nonlinearities are important and describe the dynamics of solitary-wave propagation through optical fibers and other forms of waveguides, and contains, under different circumstances, several integrable NLS-types such as the Hirota equation [43], the Sasa Satsuma model [44], Gerdjikov-Ivanov equation, Lakshmanan-Porsezian-Daniel model, Schrödinger-Hirota equation, and a variety of other such models. More specially, Eq. (1) with b 2 ¼ σ ¼ δ ¼ λ ¼ θ ¼ 0 and 0 < n < 2 was used to study chaotic motions for the perturbed NLS equation with the power-law nonlinearity based on the equilibrium points by Yin et al. [20] and was also considered by Savescu et al. [45] to analyze nonlinear dynamical problems in the nano-optical fibers. Here, we study the model Eq. (1) with arbitrary parameters that are valid for several types of highly nonlinear mediums and give rise to some new results. For this purpose, we would like to obtain the exact solutions of Eq. (1) by using the auxiliary equation method [46][47][48] and the bifurcation theory of planar dynamical systems [49,50].
The auxiliary equation method is a powerful solution method for the computation of exact traveling wave and soliton solutions. It is a one of the most direct and effective algebraic methods for finding exact solutions of nonlinear partial differential equations. This method is applicable to a large class of equations and does not need therefore to make strong assumptions about the nonlinear equations, as compared to the well-known inverse scattering transform, which uses powerful analytical methods and therefore makes strong assumptions.
The bifurcation theory of planar dynamical systems plays a crucial role in the study of the evolution of higher-order nonlinear equations. The bifurcation analysis can be used to obtain chaotic motions for Eq. (1) based on the equilibrium points. We will discuss model Eq. (1) and explore the dynamics of traveling wave solutions by employing the auxiliary equation method. In addition, using the linear stability analysis formulation, we will analyze and report the typical outcomes of the nonlinear development of the modulational instability (MI). Finally, we will point out the development of the chaotic motions for systems described by Eq. (1) through the bifurcation theory.

The auxiliary equation method
In order to obtain the exact analytic traveling wave solutions of Eq. (1), we can employ the auxiliary equation method by considering the following transformation: Here, k 1 and k 2 are real constants, U ζ ð Þ denotes the amplitude and ϕ ζ ð Þ characterizes the phase component of the soliton.
Putting Eq. (2) into Eq. (1) and separating the real and imaginary parts, one obtains and We set: The substitution of Eq. (5) into Eq. (4) gives under the restraint relation Plugging Eq. (5) into Eq. (3) with respect to Eqs. (6) and (7), we get The substitution of in Eq. (8) yields: We consider the trial equation as [46,47]: where μ l l ¼ 0, 1, … , N ð Þ are constants to be determined according to the balance principle. The previous Eq. (11) can be rewritten by the integral form Balancing VV ζζ and V 5 in Eq. (10), we get N ¼ 5. Using the solution procedure of the trial equation method [46,47], a system of algebraic equations is obtained (see Appendix) and the resolution of this obtained system yields the following: Now, from Eqs. (11) and (12), we can write The integral Eq. (14) admits many types of solutions that can be listed in accordance with [51][52][53].
, and in order to highlight our analysis, we consider the following parametric setting γ ¼ Àσ ¼ Àθ. After these considerations, the phase component of the soliton can be written as follows: where c is a real constant number.

Stability analysis method: Modulational instability (MI) of the continuous wave (CW) background
Modulational instability (MI) is a fundamental and ubiquitous phenomenon originating from the interplay between nonlinear self-interaction of wave fields and linear dispersion or diffraction. This process appears in most nonlinear systems [42,[52][53][54][55][56][57]. Unlike the well-known pulse kinds, the solitons are relatively stable, even in a perturbed environment.
In this section, we investigate the stability of the previous solutions that are sitting on a CW background, which may be subject to MI. To do so, we apply the standard linear stability analysis [52,53,56] on a generic CW (steady-state solution) where P 0 , c 1 , and c 2 are real constants. Putting Eq. (18) into Eq. (1), we get: Adding infinitesimal perturbation field υ on CW solutions by introducing the following expansionψ one can find the linearized equation satisfied by the complex perturbation υ as: where The solution of Eq. (21) is given by collecting the Fourier modes as where Ω accounts for the wavenumber and K represents the frequency of perturbation. υ þ and υ À are much less than the background amplitude P 0 (|υ z, x ð Þ| ≪ P 0 ). In this case, the instability of the steady state (CW) is achieved by the exponential growth of the perturbed field.
Deputing the expression of perturbed nonlinear background Eq. (23) into Eq. (21), we obtain after linearization, a system of homogeneous equations satisfied by υ þ and υ À : This set has a nontrivial solution only when the previous 2 Â 2 determinant matrix vanishes. By requiring the determinant of the associated matrix to be zero, we get the dispersion relation: In order to observe MI, one of the two roots of the previous dispersion relation should possess a negative imaginary part, which corresponds to an exponential growth of the perturbation amplitude. So, the MI is measured by power gain, and it is defined at any pump frequency as [52,53]: where the factor 2 converts g Ω ð Þ to power gain, and Im K max ð Þdenotes the imaginary part of the polynomial root with the largest value K max . Figure 1 depicts the MI gain spectra as a function of the modulation frequency (Ω) and second-order dispersion (a 1 ), for a fixed value of the initial power (P 0 ¼ 10kW), the other parameter values being a 2 ¼ 0: Firstly, we consider the strength of the powerlaw nonlinearity n ¼ 1, which yields to symmetrical sidelobes of instability around the zero-perturbation frequency Ω ¼ 0 in Figure 1(a). In this map, the width and magnitude of the two sidelobes remain constant in the normal group velocity dispersion (a 1 < 0), while they increase in the anomalous dispersion regime (a 1 > 0). For the nonlinearity power index n ¼ 2, we obtain in Figure 1(b), two similar sidelobes due to MI, which stand symmetrically around the line Ω ¼ 0. In this case, the magnitude and width of the sidelobes remain constant in the normal as well as the anomalous dispersion regimes. Figure 2 exhibits the enlarged MI gain spectra in 2D-plot at four power levels (P 0 ¼ 10kW; 20kW; 30kW; 50kW) with the same values of parameters as in Figure 1. The MI gain profile, indicated by colored solid curves, is constitutive of two gain bands in the Stokes frequency shift region (Ω < 0) and in the anti-Stokes frequency shift region (Ω > 0). We can see that the MI gain exists only within a limited range of frequency |Ω| < 100 ð Þand the maximum gain increases with the increasing input power P 0 .

Planar dynamical system and Hamiltonian: Phase portraits
In this section, we transform Eq. (8) to a dynamical system by introducing new variables X and Y, in order to investigate the equilibrium points, the periodic, quasiperiodic, and chaotic motions of systems in the presence of an external periodic perturbation, via the bifurcation method [49,50,58].

Formation of a dynamical system
Now, we rewrite Eq. (8) as where By setting we can rewrite Eq. (27) as a planar dynamical system The Hamiltonian of the dynamical system Eq. (30) is defined as and satisfy to This result implies that the Hamiltonian is a constant of motion [i.e., H X, Y ð Þ Cst] and the system Eq. (30) is an integrable Hamiltonian system.

Equilibrium state derivation
Using the bifurcation analysis and qualitative theory, we analyze equilibrium points for system Eq. (30). We consider the following Jacobian matrix of system (30) at the equilibrium points X k , Y k ð Þ: where X k are the zeros (solutions) of f X ð Þ, and the determinant of J is expressed by M det J ð Þ as We research equilibrium points that satisfy It is obvious to notice that Trace J ð Þ ¼ 0, and through the bifurcation theory [49,50,58], we know that, the solution X j , Y j À Á of Eq. (35) is a: From Eq. (35), firstly, we have one equilibrium point X, Y ð Þ¼ 0, 0 ð Þfor the dynamical system Eq. (30), which is a center point, and hence stable.
and by setting Δ ¼ C 2 À 4BD, we can discuss the following situations: • If Δ < 0, Eq. (36) has only one real root, which indicates that the dynamical system Eq. (30) has one equilibrium point X, Y ð Þ¼ 0, 0 ð Þ, which is a center point.
• For A 6 ¼ 0, we get We consider an evident root of the polynomial equation After this consideration, we can discuss the equilibrium points for the dynamical system Eq. (30): • If Δ 0 < 0, we get a first equilibrium point as X 0 , 0 ð Þ, which is a center point. • If Δ 0 > 0, C > 0, and D > 0, the dynamical system Eq. (30) has just equilibrium point X 0 , 0 ð Þ, which indicates a stable center point. Figure 3 shows the phase portrait of the dynamical system (30) for a 1 ¼ 1:4, , k 2 ¼ 3:5, and n ¼ 1. We observe one limit cycle about the origin 0, 0 ð Þ. This implies that the waves are stable, and there are no noises to disturb them.
The periodicity of X and Y, based on system (30) with the same values of parameters as in Figure 3, is shown in Figure 4.

Quasiperiodic and chaotic motions of the perturbed system
In this section, we will study the quasiperiodic and chaotic motions for Eq. (1) under the external perturbation. As in the previous process, we find the following perturbed dynamical system: Phase portrait of the system (30) for parameter values: where X, Y, A, B, C, and D are given by Eqs. (28) and (29); R ζ ð Þ ¼ R 0 cos ωζ ð Þ is an external periodic perturbation, R 0 is a strength of the external perturbation, and ω is the frequency. The difference between the system (30) and the system (38) is that only external periodic perturbation is added with the system (38).
In Figure 5, we have presented a phase portrait of the perturbed system (38) under the conditions of parameter values as those in Figure 3, except those fixed E 0 ¼ 0:01 and ω ¼ 1. For the same parameter values as Figure 5, we plotted in Figure 6 the quasiperiodicity of X and Y versus ζ. From this plot, it is obvious to notice that the perturbed system (38) has quasiperiodic motion even with the consideration of the external periodic perturbation.
If we increase the strength of the periodic perturbation by considering E 0 ¼ 0:01, the other parameter values remain as in Figure 5, the perturbed system (38) shows quasiperiodic route to chaos as it is shown in Figure 7. In this case, the solutions ignore the periodic motions and represent random sequences of uncorrelated oscillations (see Figures 7 and 8).  Phase portrait of the perturbed system (38) for the same parameter values as those in Figure 3, under external perturbation R ζ ð Þ ¼ E 0 cos ωζ ð Þ, where E 0 ¼ 0:01 and ω ¼ 1.
From the above observations, it is straightforward to notice that the strength of the periodic perturbation significantly enhances the development of the quasiperiodic motion of the perturbed system (38) and quasiperiodic route to chaotic motion of the system (38). Thus, the perturbed NLS Eq. (1) with the power-law nonlinearity in a nano-optical fiber not only has solitonic and periodic wave solutions but could also possess quasiperiodic and chaotic motions. Variation of X and Y with respect to ζ of the perturbed system (38), for the same values of parameters as in Figure 5.

Figure 7.
Phase portrait of the perturbed system (38) for the same parameter values as those in Figure 5 with E 0 ¼ 0:1.

Conclusion
In this chapter, we have investigated the perturbed nonlinear Schrodinger equation involving power-law nonlinearity and higher-order dispersions. We have constructed exact traveling wave solutions of the model by means of the well-known auxiliary equation method. We showed the existence of a family of traveling wave solutions and we have reported the parametric conditions on the physical parameters for the existence of these propagating solutions. Moreover, by employing Stuart and DiPrima's stability analysis method, a dispersion relation for the MI gain has been obtained. The outcomes of the instability development depend upon the nonlinearity, the power levels, and the dispersion parameters; the instability region increases regardless of the dispersion regime. The results may find straightforward applications in nonlinear optics, particularly in fiber-optical communication. Afterward, equivalent two-dimensional planar dynamic system and Hamiltonian have been derived and equilibrium points of the corresponding system have been gotten through the bifurcation theory. In addition, we have addressed the periodic, quasiperiodic, and chaotic behaviors of the traveling waves considering an external periodic perturbation. It has been observed that the perturbed system shows quasiperiodic route to chaos as a result of the strength of the periodic perturbation enhancement.