Modulation Instability Analysis of a Nonautonomous (3+1)-Dimensional Coupled Nonlinear Schrödinger Equation


 We investigate the modulation instability (MI) analysis of a nonautonomous (3+1)-dimensional coupled nonlinear Schrödinger (NLS) equation with time-dependent dispersion and phase modulation coefficients. By employing standard linear stability analysis, we obtain an explicit expression for the MI gain as a function of dispersion, phase modulation, perturbation wave numbers and an initial incidence power. The nonautonomous coupled NLS equation is found to be modulationally unstable for the same sign of dispersion and phase modulation coefficients. This equation is modulationally stable for zero dispersion and or phase modulation coefficients. But non-zero dispersion coefficient, it is modulationally stable/unstable on distinct bandwidth of wave numbers. The trigonometric, exponential, algebraic function of time and constant have been chosen as test functions for dispersion and phase modulation to find the effect on the MI analysis. The effect of focusing and defocusing medium on the MI analysis has also been investigated. The MI bandwidth in the focusing medium is found to be larger than defocusing medium. It is found that the modulation instability of the equation can be managed by proper choice of the dispersion and phase modulation parameters.


Introduction
Modulation instability (MI) is one of the critical processes that initiate the formation of periodic structures in nonlinear dispersive media. In this mechanism, a background carrier wave is modulated due to periodic seeding perturbations that initially grow but may also decay. New frequencies are generated during this process, and intensive energy exchange occurs between the participating waves. The MI process can produce an unlimited number of new frequency components, resulting in a periodic train of localized waves or pulses [1]. This effect has widespread manifestations in natural phenomena and applications in technology. The MI has also been studied in optical fibers [2,3,4], hydrodynamics [5], plasmas [6] and biology [7]. In nonlinear optics, MI has been the subject of extensive study because of its inherent connection with short pulse train generation in nonlinear optical media, as well as optical parametric amplification. The underlying mechanism involved in the initial stage of supercontinuum generation is closely connected to noise-driven MI. It has been revealed that MI is responsible for the emergence of an optical rogue wave in the supercontinuum generation process. There has been an increasing interest in the study of the variable-coefficient nonlinear evolution equations in recent years. Most of the real nonlinear wave equations possess variable coefficients that are more realistic in physical situations than their constant-coefficient counterparts. This is because the constant-coefficient models can only describe the propagation of wave groups in perfect systems. The year 1973 witnessed the first theoretical observation on optical solitons by Hasegawa and Tappert [8,9] and its experimental verification came in 1980 [10]. The mathematical description of the above observation is given by the nonlinear Schrödinger equation (NLS) equation as follows: where u = u(t, x) represents a slowly-varying electric field and x, t stand for the spatial and temporal variables, respectively. Here, u t , u xx and |u| 2 u are the time evolution, group velocity dispersion and self-phase modulation terms. σ is the self-phase modulation coefficient. Eq. (1) physically illustrates the pulse propagation in the polarization-preserving nonlinear optical fiber. Moreover, Eq. (1) is essential for studying the evolution of water waves and other nonlinear waves. A single nonlinear equation cannot appropriately describe the behaviour of solitons in optical fiber. In the present work, therefore, we have chosen a nonautonomous (3+1)dimensional coupled NLS equation [11] iu t + A(t)(u xx + u yy + u zz ) + B(t)(|u| 2 + |v| 2 )u = 0, where u = u(x, y, z, t) and v = v(x, y, z, t) represent the complex amplitude of circularly polarised waves, A(t) and B(t) are dispersion and phase management coefficients, respectively. The terms self-phase modulation and cross-phase modulation are represented by |u| 2 u, |v| 2 v and |v| 2 u, |u| 2 v, respectively. The nonlinearities |u| 2 and |v| 2 and dispersions u xx , u yy , u zz , v xx , v yy and v zz that are present in the Eq. (2) play a vital role in affecting the amplitude and phase modulation of a continuous wave. System (2) physically represents the transmission of coupled wave packets and "optical solitons" in nonlinear optical fibers and also describes transverse effects in nonlinear optical systems [12]. The closedform soliton solutions of a (3+1)-dimensional coupled NLS equation was constructed by Huang et al. [13] and Lan [14] for constant coefficients and Yu et al. [15] for time-dependent variable coefficients by Hirota method. Recently, Kumar and Patel [16] obtained dispersion and phase-managed optical soliton solutions of the Eq. (2) by complex amplitude ansatz and He's semiinverse method. The role of a closed-form solution or a numerical approximation may be replaced by MI analysis. The MI analysis has substantial significance in the study of numerical techniques employed in solving the solution of the nonautonomous (3+1)-dimensional coupled NLS equation. The procedure begins with forming an equilibrium state and ends with capturing the modulation instability gain spectrum. During the process, reasonable differential equations are drowned by introducing suitable perturbations in the equilibrium state. The non-linear dispersion relation thus obtained is solved to arrive at the final step of the MI gain for the nonautonomous (3+1)-dimensional coupled NLS equation.
The main objective of the present work is to investigate the MI analysis of the nonautonomous (3+1)dimensional coupled NLS equation by using the standard linear stability analysis. The effect of dispersion and phase modulation coefficients on the MI analysis of the equation has been studied for different choices of test functions. The dependence of the MI gain on the dispersion and phase management coefficients, perturbation wave numbers and initial incidence power has also been investigated.

Modulation instability analysis
Modulation instability (MI) refers to the exponential growth of certain modulation sidebands of the nonlinear plane waves propagating in a dispersive medium due to the interplay between nonlinearity and dispersion. In the MI process, the perturbation of the continuous wave generates the short pulses due to the breakup of the wave into a periodic pulse train. The MI of a plane wave in a nonautonomous (3+1)-dimensional coupled NLS equation is investigated by studying the stability of its amplitude in the presence of sufficiently small perturbation so that one can linearize the equation of the envelope and the carrier wave. The study of the particular aspects of these excitations in a nonautonomous (3+1)-dimensional coupled NLS equation is done by introducing a small perturbation in the amplitude and the phase and then investigate the solution of the equation (2).
For modulation instability analysis, we consider an equilibrium state solution of the nonautonomous (3+1)dimensional coupled NLS equation (2) as: where U 0 and V 0 are the real constant-amplitude perturbations. Using equilibrium state solution (3) into Eq.
(2), we obtain the time-dependent nonlinear phase shift φ 1 (t) and φ 2 (t) as This result shows that the nonlinear phase shift φ 1 (t) and φ 2 (t) of the equilibrium state solutions are found to be the function of phase modulation coefficient B(t) only. Now, we introduce small perturbations in Eq. (3) to perform the stability analysis of the steady-state solutions, giving rise to the following In Eq.(4), U 1 and V 1 represent the complex amplitude of the perturbation, |U 1 (x, y, z, t)| U 0 and |V 1 (x, y, z, t)| V 0 . Substituting Eq. (4) in Eq. (2) and linearizing the equation with respect to U 1 , U * 1 and V 1 , V * 1 , where the asterisk stands for the corresponding complex conjugate, we obtain the evolution equations for the perturbations as The Eq. (5) is a coupled linear partial differential equation. Therefore, we assume the solution of Eq. (5) in the form where r 0 (t), r 1 (t), s 0 (t), s 1 (t) and Λ(t) are real valued functions of t. The term Kx+Ly+M z − t 0 Λ(t )dt represents the phase of modulation, K, L and M are the perturbation wave numbers in x, y and z directions and Λ(t) is the time-dependent perturbation wave frequency of the modulation waves. Substituting Eqs. (6) into (5) and separating the coefficients of we get the following timedependent linear homogeneous equations in r 0 (t), r 1 (t), s 0 (t), and s 1 (t) as where Equating the real and imaginary parts in Eq. (7), we get r 0t = r 1t = s 0t = s 1t = 0. Further, the obtained system of equations can be represented in matrix form as In order to obtain a nontrivial solution, the determinant of the coefficient matrix may be allow to vanish, which gives the time-dependent dispersion relation This relation reveals the dependency of the time-dependent oscillations e iΛ(t) on the spatial oscillations e iKx , e iLy , and e iM z . The stability of the perturbed steady-state solution depends on the real or complex value of Λ(t).
For B(t) = 0, the Λ(t) is always real and modulation instability does not occur. Similarly, for A(t) = 0, the Λ(t) = 0, so the system is again modulationally stable. The steady-state solution becomes unstable whenever Λ(t) has an imaginary part because, in this case, perturbation grows exponentially along the fiber length. This phenomenon is called modulation instability [4,31,32] as it leads to modulation of the steady-state solution. From Eq. (9), we can observe that the steady-state solution of a nonautonomous (3+1)-dimensional coupled NLS equation will be modulationally stable or unstable whenever ≥ 0 or < 0, respectively. It has been deduced that Λ(t) is imaginary when either A(t) and B(t) both positive or both negative. The MI gain G of the nonautonomous (3+1)-dimensional coupled NLS equation could be written as .
Clearly, the Λ(t) depends on dispersion coefficient A(t) and phase modulation coefficient B(t), perturbation wave numbers K, L and M , and incidence power U 0 and V 0 in equilibrium state solutions. For A(t) = 0, the gain G = 0, for the wave number − (L 2 + M 2 ) and the coupled NLS equation is modulationally stable for bandwidth |K| ≥ |K s | and unstable for instability bandwidth |K| < |K s |. The width of the instability band depends on the dispersion and phase managed coefficients, incidence power (U 0 , V 0 ) and perturbation wave numbers L and M . The MI gain attains a minimum value , at wave number K = 0, and maximum value . It is found that K or/ L and G min can be controlled by dispersion and phase modulation both but G max can only be controlled by phase managed coefficient. Further, G max is independent from the wave numbers L and M. The gain spectra G is plotted against wave number K in Figs

Results and Discussion
The stability of the steady-state solutions of the nonautonomous(3+1)-dimensional coupled NLS equation has been investigated by the method of the modulation instability (MI) analysis. The MI gain G is calculated and plotted against time t, and the perturbation wave numbers K and L for a fixed value of M , U 0 , and V 0 . From equation (10), we have found that the equilibrium state (U 0 , V 0 ), normalized wave numbers K, L, and M , dispersion A(t) and phase modulation B(t) have a significant effect on the MI gain G. In the absence of dispersion (A(t) = 0) and/or phase modulation (B(t) = 0) the system is found to be modulationally stable. It has also been found that the MI gain exists only when either A(t) > 0, B(t) > 0 or A(t) < 0, B(t) < 0. This observation is similar to the finding of [4] for constant coefficients. The effects of the distinct time-dependent dispersion and phase modulation coefficients on the MI gain have been discussed under the following subsections: 100 (see Fig.  2a), the MI gain witnessed dual-band, which depicts that it is independent of the sign of the K. There exist two local maxima in the MI gain and an instability band for wave number K for a fixed value of the initial incidence power (U 0 , V 0 ). With an increase in the initial incidence power (U 0 , V 0 ), the local maxima and the width of the instability band both increase. The contour plot of the MI has been shown in Figure 1c. Figs. 2b-2d, gives the variation in the MI gain with the change in values of perturbation wave numbers L and M . With an increase in the value of incidence power (U 0 , V 0 ), there develops a single MI band for L = 0.2 and M = 0.5 (see Fig. 2b). For U 0 = 1.0 and V 0 = 2.0, there exist a local MI gain maxima at K = 0. But for U 0 = 1.5 and V 0 = 2.5, the MI gain achieve the maxima in the interval |K| < 0.2. With a decrease in the value of L and M , the single MI band (see Fig. 2b) split into two fully developed sidebands (see Fig. 2d). The partially developed sidebands are shown in Fig. 2c. The width of the instability band for wave number K increases with the decreases in the perturbation wave number L and M . For L = 0.2, M = 0, the MI gain G attains local minima at K = 0 (see Fig. 2c). For L = 0, M = 0, the local minima in the MI gain G hits the zero gain at K = 0 (see Fig. 2d). Figure 3 shows that the MI gain and instability bandwidth increases with the increase in initial incidence power but decreases with time. The contour plot (see Figs. 3b and 3d) also confirm it.

Impact of trigonometric and exponential dispersion and phase modulation on MI
The surface plots of the MI gain (see Figs. 4a-4b) for A(t) = sint and B(t) = − exp(t) for π < t < 2π show a similar pattern as in Figs. 1a-1b, but the gain increases exponentially with time t. For t = 3π 2 , Fig. 2a depicts the two-dimensional variation of the MI gain. For A(t) = − exp(t) and B(t) = sint, the two-dimensional plot of the MI gain is given in Figs. 5b-5d. The two local maxima in the MI gain and MI bandwidth increases with phase-managed coefficient B(t) = − exp(t) (see Fig. 5a), keeping a similar pattern of the MI gain spectrum for B(t) = sint (see Fig. 2a). The contour plot of the MI gain has been shown in Figure 4c. Fig.  5b gives the zero MI gain for all the values of K, opposite to the case of a single band of Fig. 2b for corresponding values of L and M and initial incidence power (U 0 , V 0 ). It is found that for initial incidence power  (U 0 , V 0 ) > (2.6, 2.9), the single MI gain band originates. Therefore, we may conclude that the dispersion coefficient A(t) and incidence power (U 0 , V 0 ) largely affects the MI gain of the coupled NLS equation. Figs. 5c-5d show a similar pattern of the MI gain as in Figs. 2c-2d with higher values of local maxima and MI bandwidth. Figures 3 and 6 show that MI gain and instability bandwidth increases significantly with the change in phase modulation. Further, it is seen that MI gain and instability bandwidth both are symmetric with respect to wave numbers.  For the corresponding value of incidence power (U 0 , V 0 ), the local maxima in both the focusing and defocusing cases are equal, but the MI bandwidth is quite larger in the focusing case. In focusing medium (B(t) = +1), the MI gain and instability bandwidth both decrease with time (see Fig. 9). In defocusing medium, the gain is not changing significantly, but instability bandwidth decreases for small change in time (see Fig. 11). Also, in the neighbourhood of time t = π, the MI gain and instability bandwidth are similar in focusing and defocusing medium (see Figs. 9c and 11c)

Comparative analysis:
We have compared our result of the MI analysis of the nonautonomous (3+1)-dimensional coupled nonlinear Schrödinger equation with [36] and [37]. We have a   equilibrium state solution only in x−direction. We have found that the MI gain spectrum obtained in Fig. (12) are very similar to the spectrum obtained in [37] and better in comparison to the result in [36]. This validates the extension of the MI analysis for the case of a nonau-tonomous (3+1)-dimensional coupled NLS equation in this work.

Conclusions
This paper investigates the modulation instability of a nonautonomous (3+1)-dimensional coupled NLS equation. We have considered small perturbations in steadystate solution in all the directions x, y, and z in terms of perturbation wave numbers. It is found that the non-linear phase shift in steady-state solution depends only on the phase modulation coefficient. Some important observations about the modulation instability of a nonautonomous (3+1)-dimensional coupled NLS equation can be summarised as follows: -The MI gain is found to be a function of time t, the initial incidence power and perturbation wave numbers. The gain is symmetric with respect to the wave numbers. -The system of nonautonomous coupled NLS equation is modulationlly stable in the absence of dispersion and/or phase modulation against weak perturbations. The cross-phase modulation, together with self-phase modulation, is responsible for novel instability. -The MI gain exists only for the same sign of dispersion and phase modulation. It can be controlled by choosing desired test function for dispersion and phase modulation coefficients. -The width of the instability band depends on the dispersion and phase modulation coefficients in coupled NLS equation, incidence power and perturbation wave numbers in the steady-state solution. -The value of the MI gain for the exponential coefficient is much higher than the trigonometric coefficient for fixed values of the other parameters. For algebraic and constant dispersion and phase modulation, the MI bandwidth is independent of perturbation wave numbers. -The local maxima in the MI gain in the focusing and defocusing medium is equal but the MI bandwidth is found significantly larger for the focusing medium.
The MI analysis of a nonautonomous (3+1)-dimensional coupled NLS equation reveals that the modulation instability of the equation can be controlled by the proper choice of the dispersion and phase modulation coefficients, initial incident power and perturbation wave numbers in three dimensions. Although this study is limited in the context of optical fiber but the analysis may find application in other nonlinear dispersive system and in different branches of physics.