Rational Waves and Complex Dynamics : Analytical Insights into a Generalized Nonlinear Schrödinger Equation with Distributed Coefficients

In this paper, we first present a complex multirational exp-function ansatz for constructing explicit solitary wave solutions, Nwave solutions, and rouge wave solutions of nonlinear partial differential equations (PDEs) with complex coefficients. To illustrate the effectiveness of the complex multirational exp-function ansatz, we then consider a generalized nonlinear Schrödinger (gNLS) equationwith distributed coefficients. As a result, some explicit rational exp-function solutions are obtained, including solitarywave solutions, N-wave solutions, and rouge wave solutions. Finally, we simulate some spatial structures and dynamical evolutions of the modules of the obtained solutions for more insights into these complex rational waves. It is shown that the complex multirational exp-function ansatz can be used for explicit solitary wave solutions, N-wave solutions, and rouge wave solutions of some other nonlinear PDEs with complex coefficients.


Introduction
In the real world, complex nonlinear phenomena are everywhere and nonlinear PDEs are often used to describe these nonlinear complexities.To gain more insights into the essence behind the nonlinear phenomena for further applications, people usually restore to the dynamical evolutions of exact wave solutions of nonlinear PDEs.It is well known that the celebrated Schrödinger wave equation possesses Nsoliton solutions and is often used to describe quantum mechanical behavior.In the field of nonlinear mathematical physics, many analytical methods have been presented for exactly solving nonlinear PDEs, such as those in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].It is worth mentioning that the exp-function method [8] with a rational exp-function ansatz is an effective mathematical tool for constructing exact wave solutions.
In this paper, with a complex multirational exp-function ansatz, we shall construct and gain more insights into the rational solutions, including solitary wave solutions, N-wave solutions, and rouge wave solutions of the following gNLS equation with gain in the form used in nonlinear fiber optics [20][21][22][23][24]: where  = (, ) is a complex-valued function of the propagation distance  and the retarded time , while (), (), and () are all differentiable functions of , which denote the group velocity dispersion, nonlinearity, and distributed gain, respectively.If we set then (1) can be reduced to the well-known NLS equation: The rest of the paper is organized as follows.In Section 2, we give a description of the complex multirational expfunction ansatz used to construct explicit solitary wave solutions, N-wave solutions, and rouge wave solutions of nonlinear PDEs with complex coefficients.In Section 3, we use the introduced complex multirational exp-function ansatz to construct solitary wave solutions, N-wave solutions, and rouge wave solutions of the gNLS in (1).In Section 4, in order to gain more insights into the complex dynamics of the obtained wave solutions, we simulate the dynamical evolutions of some solitary wave solutions, N-wave solutions, and rouge wave solutions.In Section 5, we conclude this paper.

Complex Multirational Exp-Function Ansatz
For a given nonlinear PDE with complex coefficients, for example, the NLS in (3), we suppose that its complex multirational exp-function ansatz has the following form [9]: where , and   are complex constants to be determined by substituting (4) into (3),   is an arbitrary complex constant, the real values of  1 ,  2 , ⋅ ⋅ ⋅ ,  2 ,  1 ,  2 , ⋅ ⋅ ⋅ ,  2 are integers determined by the process of homogeneous balance, and * denotes the complex conjugate.
Special case 1 of (4): solitary wave ansatz: for the separate real and imaginary parts of the NLS in (3) in an indirect way.Special case 2 of (4): N-wave ansatz: When  = 1, (4) gives which can be used to construct single-wave solution of the NLS in (3) in a direct way.When  = 2, (4) gives which can be used to construct double-wave solution of the NLS in (3) in a direct way.
When  = 3, (4) gives which can be used for three-wave solution of the NLS in (3) in a direct way.Special case 3 of (4): rouge wave ansatz: with the constraints  1 = − * 1 and  1 =  * 1 , which can be used for the NLS in (3) in a direct way.

Rational Exp-Function Solutions
In this section, we employ the rational exp-function ansatz (4) and its special cases (5)-( 9) to construct rational solutions, including solitary wave solutions, N-wave solutions, and rouge wave solutions of the gNLS in (1).
We therefore obtain a pair of rational exp-function solutions of the gNLS in (1): where

N-Wave Solutions.
For convenience, we let and we still write  as ; then (1) is converted into In what follows, we construct N-wave solutions of ( 23).To begin with the single-wave solution, we suppose that Substituting ( 24) into ( 23) and equating each coefficient of the same order power of e  1 + * 1 (,  = 0, 1, 2, ⋅ ⋅ ⋅ ) to zero yield a set of algebraic equations for  0 ,  1 ,  2 ,  3 ,  1 ,  2 ,  3 ,  1 , and  1 .Solving the set of equations, we have and hence we obtain the single-wave solution: where  1 ,  1 , and  1 are arbitrary complex constants and For the double-wave solution, we next suppose that where 23) and equating each coefficient of the same order 4 Complexity power of e  1 + 2 + * 1 + * 2 (, , ,  = 0, 1, 2, ⋅ ⋅ ⋅ ) to zero yield a set of algebraic equations, from which we have and hence we obtain the double-wave solution as follows: where and , and  2 are arbitrary complex constants.
Finally, we determine the three-wave solution of the following form: where Similarly, we have Complexity With the help of ( 40)-( 61), the three-wave solution (39) can be finally determined as follows: where and the summation Σ =0,1 refers to all the combinations of   = 0, 1( = 1, 2, 3);  1 () and  2 () denote that all the following conditions must hold: Generally, introducing the notations we can obtain a uniform formula of the N-wave solution: where the summation Σ =0,1 refers to all the combinations of   = 0, 1( = 1, 2, ⋅ ⋅ ⋅ , );  1 () and  2 () denote that all the following conditions must hold: (75)

Rouge Wave Solutions.
To construct rouge wave solutions, we rewrite (9) as and we substitute (76) into ( 23); then we equate each coefficient of the same order power of cos  e (+) (,  = 0, 1, 2, ⋅ ⋅ ⋅ ) to zero; a set of algebraic equations is derived.Solving the set of equations, we have and We, therefore, obtain two pairs of rational exp-function wave solutions as follows: It is easy to see that when  2 = 1,  = 0, and  < 0 the molecular and denominator of solution (81) tend to zeros, respectively.We differentiate (81) with respect to  twice and let  → 0; then the limits of solution (81) give two rouge wave solutions: In a similar way, when  2 = 1,  = 0, and  > 0, the limits of solution (82) give two rouge wave solutions, which are the same as those in (83) and (84), respectively.

Complex Dynamics
To gain more insights into the solutions obtained in Section 3, we investigate the dynamical evolutions of some obtained solutions.Firstly, we select  1 = 1,  0 = 1,  0 = 0.1, ℎ 0 = 2, () =  3 , and () = −; then the modules of solution (20) with "+" branch and different values of  0 are shown in Figures 1-4, respectively.It is shown in Figures 1-4 that when the other parameters are fixed, the larger the value of  0 is, the smaller the influence on M-shape wave will be.
Secondly, we consider solutions ( 26), (33), and (62).In Figure 5, the module of the single-wave solution ( 26) is shown by selecting  1 = 1+i,  1 = 1−0.2i,and  = 1.We simulate the module of the double-wave solution (33) in Figure 6, where  1 = 0. of  has influenced the spatial structures of the module of solution (81) and can also lead to the periodicity and singularity.

Figure 20 :
Figure 20: Dynamical evolutions of the rouge wave structure determined by the module of solution (83).