Wave behaviors of Kundu–Mukherjee–Naskar model arising in optical fiber communication systems with complex structure

Rogue waves are very mysterious and extra ordinary waves. They appear suddenly even in a calm sea and are hard to be predicted. Although nonlinear Schrödinger equation provides a perspective, it alone can neither detect rogue waves nor provide a complete solution to problems. Therefore, some approximations are still mandatory for both obtaining an exact solution and predicting rogue waves. Such as Kundu–Mukherjee–Naskar (KMN) model which allows obtaining lump-soliton solutions considered as rogue waves. In this study the functional variable method is utilized to obtain the analytical solutions of KMN model that corresponds to the propagation of soliton dynamics in optical fiber communication system.


Introduction
Wave motion is very predictable in basic level and can be explained very deterministic way. However, it becomes much more complicated with taking the nonlinear interactions into account. This type of systems called nonlinear or dynamical systems. Although, there has been some development in the mathematical field to understand the behavior of a nonlinear system we are still far from predicting the behavior of such systems deterministically. The first studies on this issue date back almost 150 years to Riemann and stokes. And the studies are continuing with increasing importance and interest (Rezazadeh et al. , 2019Rezazadeh 2018). In the last decade this interest increased more than before (Kundu et al. 2021;Yang et al. 2021;Alam et al. 2021;Ma et al. 2021;Alam and Osman 2021;Gómez-Aguilar et al. 2021; Barman et al. 2021). Many different analytical methods are applied to understand and explain physical behavior of the nonlinear systems. (Barman et al. 2021;Osman et al. 2018;Liu et al. 2019a, b;Ding et al. 2019;Ekici et al. 2016;Eslami and Mirzazadeh 2016) The basic idea is concentrated on two types of wave behaviors which are hyperbolic (Pettersson et al. 2019) and dispersive. Hyperbolic wave behavior can be formulated mathematically in terms of hyperbolic partial differential equations. Klein-Gordon (Kurt 2019) is a prototype for hyperbolic wave equation. Dispersive waves cannot be characterized easily. Nonhyperbolic waves generally categorized as dispersive waves. However, classification is made on the type of solution rather than on the equations. Korteweg-DeVries equation (Kurt et al. 2017) can be a good example for dispersive waves. There are many equations developed for determining the wave behavior of a dynamical system (Gao et al. 2020;Rezazadehd et al. 2018;Raza et al. 2019a, b;Tasbozan et al. 2019;Atilgan et al. 2019;Kurt 2019;Kurt et al. 2017;Seadawy et al. , 2020Ali et al. 2018a, b;Raslan et al. 2017;Sedeeg et al. 2019;Sulaiman et al. 2019a, b;Bulut 2019, 2020).
For a small intersection of a spatiotemporal system the dynamics can be assumed as linear. However, they must be evaluated in terms nonlinear dynamics due to significant modulation of the wave amplitude originated from cumulative nonlinear interactions. Nonlinear Schrödinger (NLS) equation is a very common equation which is providing a canonical design of involucre dynamics of a quasi-monochromatic planar wave propagating in a weakly nonlinear dispersive medium when dissipative effects are insignificant (Liu et al. 2018;El-Dessoky and Islam 2019;Seadawy and Cheemaa 2019).
NLS is employed for many situational models such as propagation of a wave in a Kerr type (Zhang et al. 2011a, b) or non-Kerr type (Liu 2010) medium. Most of them are not fully integrable which means exact solutions cannot be obtained directly. Only approximate numerical solutions with no stable solitons can be obtained (Zhang and Simos 2016a, b). Approximations cannot predict rogue waves which can be defined as "localized and isolated surface waves, apparently appear from nowhere, make a sudden hole in the sea just before attaining surprisingly high amplitude and disappear again without a trace" KMN (Kundu et al. 2014). They proposed a model to by extension of NLS to have an integrable form which allows lump-soliton can be considered as rogue wave model; and then, they replaced the conventional amplitude-like nonlinear term with the a currentlike nonlinear term which allows them to obtain a fully integrable form of NLS; In this study the wave solutions of KMN model which describes the propagation of soliton dynamics in optical fiber communication system. Yıldırım (Yıldırım 2019) obtained dark, bright and singular solitons by using trial equation technique for KMN model. Rivzi et al. (Rizvi et al. 2020) used csch method, extended Tanh-Coth method and extended rational sinh-cosh method to get the exact solutions of KMN model. Talarposhti et al. (Talarposhti et al. 2020) employed Exp-function method to yield the optical soliton solutions of considered KMN model. This work is structured as follows: In Sect. 2, mathematical analysis of KMN model is given. In Sect. 3, we demonstrate the structure of the functional variable method. In Sect. 4, we apply this method to find some wave solutions of the equation written above. In Sect. 5, we give the results and discussion, Sect. 6 gives the conclusion of the whole research.

Mathematical analysis
In order to get started, the following hypothesis is selected: where P(ξ) represents the amplitude portion and and the phase portion of the soliton is defined as Here, 1 and 2 are the frequencies of the soliton in the x-and y-directions respectively while is the wave number of the soliton and finally 0 is the phase constant. Also, the parameters 1 and 2 in (2.2) represent the inverse width of the soliton along x-and y-directions respectively, while 2.3 represents the velocity of the soliton. Inserting (2.1) along with (2.2) and (2.3) into (1.1) and decomposing into real and imaginary parts, the following pair of equations, respectively yield Equation (2.4) is transformed into the following one (2.4) 1 2 P �� − ( + 1 2 )P − 2 1 P 3 = 0, (2.5) = − ( 1 2 + 2 1 ).

The functional variable method
This section presents the brief descriptions of the functional variable method (Zerarka and Ouamane 2010;Eslami et al. 2017;Bekir et al. 2015). While applying this method discretization or normalization is not needed, this is the main advantage of the method. Also, nonlinear partial differential equation is converted into nonlinear ordinary differential equation by the help of wave transform and chain rule. This process makes the solution easier and faster. Suppose that a the NLEE, say in two independent variables to x and t is given by where G is a function of u, u t , u xx , … and the subscripts denote the partial derivatives of u(x, t) with respect to x and t. A transformation u(x, t) = U( ), = x − t converts the NLEE (3.1) to a nonlinear ODE where F is a function of U, U , U , … and its derivatives point out the ordinary derivatives with respect to and where and is constant to be determine.
Then we make a transformation in which the unknown function U is considered as a functional variable in the form: and some successive derivatives of U are where ''′" stands for d dU . The ODE (3.2) can be reduced in terms of U, F3.4 and its derivatives upon using the expressions of Eq. (3.4) into Eq. (3.2) gives by integrating of Eq. (3.5), Eq. (3.5) can be written with respect to H , and it is found the appropriate solutions by using Eq. (3.3) for the investigated problem.
(4.2) Ω 2 = + 1 2 1 2 (4.5) q ± 2 (x, y, t) = ± √ + 1 2 1 csch √ + 1 2 1 2 1 x + 2 y + ( ( 1 2 + 2 1 ))t , we obtain the following periodic wave solutions 5 Results and discussion Figure 1 shows the graphs obtained from the space-time mapping of the solution q 1 . It can be seen from the figure that the waves have a spatiotemporally extended homoclinic breather wave structure. In this respect, it can be concluded that this q 1 solution can be useful in examining the dynamic behavior of rogue waves. It can also be seen that breather waves extend periodically along with time while extending at a certain angle with the X-axis spatially. Figure 2 shows the graphs obtained from the space-time mapping of the solution q 4 . Interestingly, although this solution seems to be the solution of heteroclinic waves at first glance, a careful look reveals the difference of the situation. This solution shows the existence of periodically extended homoclinic waves both spatially and temporally. Figure 2 shows the graphs obtained from the space-time mapping of the solution q 5 . From this solution, the existence of singular waves extended in time can be seen Fig. 3.