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

Abstract

In this paper, we first present a complex multirational exp-function ansatz for constructing explicit solitary wave solutions, N-wave 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) equation with distributed coefficients. As a result, some explicit rational exp-function solutions are obtained, including solitary wave 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.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. 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 N-soliton 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 [1–19]. 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 [20–24]: $$\begin{array}{}\text{(1)}& \mathrm{i}{\psi }_x+\frac{\mathrm{1}}{\mathrm{2}}\alpha \left(x\right){\psi }_{tt}+\beta \left(x\right){\left|\psi \right|}^{\mathrm{2}}\psi =\mathrm{i}\gamma \left(x\right)\psi ,\end{array}$$ where $\psi =\psi (x,t)$ is a complex-valued function of the propagation distance $x$ and the retarded time $t$ , while $\alpha (x)$ , $\beta (x)$ , and $\gamma (x)$ are all differentiable functions of $x$ , which denote the group velocity dispersion, nonlinearity, and distributed gain, respectively. If we set $$\begin{array}{c}\text{(2)}& \alpha \left(x\right)=\mathrm{2},\beta \left(x\right)=\pm \mathrm{1},\\ \gamma \left(x\right)=\mathrm{0},\end{array}$$ then (1) can be reduced to the well-known NLS equation: $$\begin{array}{}\text{(3)}& \mathrm{i}{\psi }_x+{\psi }_{tt}\pm {\left|\psi \right|}^{\mathrm{2}}\psi =\mathrm{0}.\end{array}$$

The rest of the paper is organized as follows. In Section 2, we give a description of the complex multirational exp-function 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.

2. 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]: $$\begin{array}{}\text{(4)}& \psi =\frac{{\sum }_{i_{\mathrm{1}}=\mathrm{0}}^{p_{\mathrm{1}}}\hspace{0.17em}{\sum }_{i_{\mathrm{2}}=\mathrm{0}}^{p_{\mathrm{2}}}\cdots {\sum }_{i_{\mathrm{2}N}=\mathrm{0}}^{p_{\mathrm{2}N}}{a}_{i_{\mathrm{1}}{i}_{\mathrm{2}}\cdots i_{\mathrm{2}N}}{\mathrm{e}}^{{\sum }_{g=\mathrm{1}}^N\left(i_g{\xi }_g+i_{g+N}{\xi }_g^{\ast }\right)}}{{\sum }_{j_{\mathrm{1}}=\mathrm{0}}^{q_{\mathrm{1}}}\hspace{0.17em}{\sum }_{j_{\mathrm{2}}=\mathrm{0}}^{q_{\mathrm{2}}}\cdots {\sum }_{j_{\mathrm{2}N}=\mathrm{0}}^{q_{\mathrm{2}N}}{b}_{j_{\mathrm{1}}{j}_{\mathrm{2}}\cdots j_{\mathrm{2}N}}{\mathrm{e}}^{{\sum }_{g=\mathrm{1}}^N\left(j_g{\xi }_g+j_{g+N}{\xi }_g^{\ast }\right)}},\hspace{1em}N\ge \mathrm{1},\end{array}$$ where ${\xi }_g=k_{g}t+c_{g}x+d_g$ , ${\xi }_g^{\ast }=k_g^{\ast }t+c_g^{\ast }x+d_g^{\ast }$ , $a_{i_{\mathrm{1}}{i}_{\mathrm{2}}\cdots i_{\mathrm{2}N}}$ , $b_{j_{\mathrm{1}}{j}_{\mathrm{2}}\cdots j_{\mathrm{2}N}}$ , $c_g$ , and $k_g$ are complex constants to be determined by substituting (4) into (3), $d_g$ is an arbitrary complex constant, the real values of $p_{\mathrm{1}},p_{\mathrm{2}},\cdots ,p_{\mathrm{2}N},q_{\mathrm{1}},q_{\mathrm{2}},\cdots ,q_{\mathrm{2}N}$ are integers determined by the process of homogeneous balance, and $\ast$ denotes the complex conjugate.

Special case 1 of (4): solitary wave ansatz: $$\begin{array}{}\text{(5)}& \psi =\frac{a_{\mathrm{0}}+a_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}+a_{\mathrm{2}}{\mathrm{e}}^{\mathrm{2}{\xi }_{\mathrm{1}}}}{b_{\mathrm{0}}+b_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}+b_{\mathrm{2}}{\mathrm{e}}^{\mathrm{2}{\xi }_{\mathrm{1}}}},\end{array}$$ 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 $N=\mathrm{1}$ , (4) gives $$\begin{array}{}\text{(6)}& \psi =\frac{{\sum }_{i_{\mathrm{1}}=\mathrm{0}}^{p_{\mathrm{1}}}\hspace{0.17em}{\sum }_{i_{\mathrm{2}}=\mathrm{0}}^{p_{\mathrm{2}}}{a}_{i_{\mathrm{1}}{i}_{\mathrm{2}}}{\mathrm{e}}^{i_{\mathrm{1}}{\xi }_{\mathrm{1}}+i_{\mathrm{2}}{\xi }_{\mathrm{1}}^{\ast }}}{{\sum }_{j_{\mathrm{1}}=\mathrm{0}}^{q_{\mathrm{1}}}\hspace{0.17em}{\sum }_{j_{\mathrm{2}}=\mathrm{0}}^{q_{\mathrm{2}}}{b}_{j_{\mathrm{1}}{j}_{\mathrm{2}}}{\mathrm{e}}^{i_{\mathrm{1}}{\xi }_{\mathrm{1}}+i_{\mathrm{2}}{\xi }_{\mathrm{1}}^{\ast }}},\end{array}$$ which can be used to construct single-wave solution of the NLS in (3) in a direct way.

When $N=\mathrm{2}$ , (4) gives $$\begin{array}{}\text{(7)}& \psi =\frac{{\sum }_{i_{\mathrm{1}}=\mathrm{0}}^{p_{\mathrm{1}}}\hspace{0.17em}{\sum }_{i_{\mathrm{2}}=\mathrm{0}}^{p_{\mathrm{2}}}\hspace{0.17em}{\sum }_{i_{\mathrm{3}}=\mathrm{0}}^{p_{\mathrm{3}}}\hspace{0.17em}{\sum }_{i_{\mathrm{4}}=\mathrm{0}}^{p_{\mathrm{4}}}{a}_{i_{\mathrm{1}}{i}_{\mathrm{2}}{i}_{\mathrm{3}}{i}_{\mathrm{4}}}{\mathrm{e}}^{{\sum }_{g=\mathrm{1}}^{\mathrm{2}}\left(i_g{\xi }_g+i_{g+\mathrm{2}}{\xi }_g^{\ast }\right)}}{{\sum }_{j_{\mathrm{1}}=\mathrm{0}}^{q_{\mathrm{1}}}\hspace{0.17em}{\sum }_{j_{\mathrm{2}}=\mathrm{0}}^{q_{\mathrm{2}}}\hspace{0.17em}{\sum }_{j_{\mathrm{3}}=\mathrm{0}}^{q_{\mathrm{3}}}\hspace{0.17em}{\sum }_{j_{\mathrm{4}}=\mathrm{0}}^{q_{\mathrm{4}}}{b}_{j_{\mathrm{1}}{j}_{\mathrm{2}}{j}_{\mathrm{3}}{j}_{\mathrm{4}}}{\mathrm{e}}^{{\sum }_{g=\mathrm{1}}^{\mathrm{2}}\left(j_g{\xi }_g+j_{g+\mathrm{2}}{\xi }_g^{\ast }\right)}},\end{array}$$ which can be used to construct double-wave solution of the NLS in (3) in a direct way.

When $N=\mathrm{3}$ , (4) gives $$\begin{array}{}\text{(8)}& \psi =\frac{{\sum }_{i_{\mathrm{1}}=\mathrm{0}}^{p_{\mathrm{1}}}\hspace{0.17em}{\sum }_{i_{\mathrm{2}}=\mathrm{0}}^{p_{\mathrm{2}}}\hspace{0.17em}{\sum }_{i_{\mathrm{3}}=\mathrm{0}}^{p_{\mathrm{3}}}\hspace{0.17em}{\sum }_{i_{\mathrm{4}}=\mathrm{0}}^{p_{\mathrm{4}}}\hspace{0.17em}{\sum }_{i_{\mathrm{5}}=\mathrm{0}}^{p_{\mathrm{5}}}\hspace{0.17em}{\sum }_{i_{\mathrm{6}}=\mathrm{0}}^{p_{\mathrm{6}}}{a}_{i_{\mathrm{1}}{i}_{\mathrm{2}}{i}_{\mathrm{3}}{i}_{\mathrm{4}}{i}_{\mathrm{5}}{i}_{\mathrm{6}}}{\mathrm{e}}^{{\sum }_{g=\mathrm{1}}^{\mathrm{3}}{i}_g\left(i_g{\xi }_g+i_{g+\mathrm{3}}{\xi }_g^{\ast }\right)}}{{\sum }_{j_{\mathrm{1}}=\mathrm{0}}^{q_{\mathrm{1}}}\hspace{0.17em}{\sum }_{j_{\mathrm{2}}=\mathrm{0}}^{q_{\mathrm{2}}}\hspace{0.17em}{\sum }_{j_{\mathrm{3}}=\mathrm{0}}^{q_{\mathrm{3}}}\hspace{0.17em}{\sum }_{j_{\mathrm{4}}=\mathrm{0}}^{q_{\mathrm{4}}}\hspace{0.17em}{\sum }_{j_{\mathrm{5}}=\mathrm{0}}^{q_{\mathrm{5}}}\hspace{0.17em}{\sum }_{j_{\mathrm{6}}=\mathrm{0}}^{q_{\mathrm{6}}}{b}_{j_{\mathrm{1}}{j}_{\mathrm{2}}{j}_{\mathrm{3}}{j}_{\mathrm{4}}{j}_{\mathrm{5}}{j}_{\mathrm{6}}}{\mathrm{e}}^{{\sum }_{g=\mathrm{1}}^{\mathrm{3}}\left(j_g{\xi }_g+j_{g+\mathrm{3}}{\xi }_g^{\ast }\right)}},\end{array}$$ 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: $$\begin{array}{}\text{(9)}& \psi =\frac{a_{\mathrm{0}}+a_{\mathrm{1}}\cos \left(\mathrm{1}/\mathrm{2}\right)\left({\xi }_{\mathrm{1}}-{\xi }_{\mathrm{1}}^{\ast }\right){\mathrm{e}}^{{\xi }_{\mathrm{1}}-k_{\mathrm{1}}t}+a_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }}}{b_{\mathrm{0}}+b_{\mathrm{1}}\cos \left(\mathrm{1}/\mathrm{2}\right)\left({\xi }_{\mathrm{1}}-{\xi }_{\mathrm{1}}^{\ast }\right){\mathrm{e}}^{{\xi }_{\mathrm{1}}-k_{\mathrm{1}}t}+b_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }}},\end{array}$$ with the constraints $k_{\mathrm{1}}=-k_{\mathrm{1}}^{\ast }$ and $c_{\mathrm{1}}=c_{\mathrm{1}}^{\ast }$ , which can be used for the NLS in (3) in a direct way.

3. 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).

3.1. Solitary Wave Solutions

Let us begin with the gNLS in (1). Firstly, we assume that $\alpha (x)$ , $\beta (x)$ , and $\gamma (x)$ are all real functions and let $$\begin{array}{c}\text{(10)}& \psi =A{\mathrm{e}}^{\mathrm{i}\theta },A=A\left(x,t\right),\\ \theta =\theta \left(x,t\right),\end{array}$$ where $A$ and $\theta$ are the amplitude and phase functions, respectively. With the help of (10), we separate the real and imaginary parts of (1) as follows: $$\begin{array}{}\text{(11)}& -A{\theta }_x+\frac{\mathrm{1}}{2}\alpha \left(x\right)\left(A_{tt}-A{\theta }_t^{\mathrm{2}}\right)+\beta \left(x\right)A^{\mathrm{3}}=\mathrm{0},\text{(12)}& A_x+\frac{\mathrm{1}}{2}\alpha \left(x\right)\left(\mathrm{2}{A}_t{\theta }_t+A{\theta }_{tt}\right)-\gamma \left(x\right)A=\mathrm{0}.\end{array}$$

Then we further suppose that $$\begin{array}{}\text{(13)}& A=\frac{a_{\mathrm{0}}\left(x\right)+a_{\mathrm{1}}\left(x\right){\mathrm{e}}^{\xi }}{\mathrm{1}+b_{\mathrm{1}}{\mathrm{e}}^{\xi }},\hspace{1em}\xi =p\left(x\right)+q\left(x\right)t,\text{(14)}& \theta =f\left(x\right)t^{\mathrm{2}}+g\left(x\right)t+h\left(x\right),\end{array}$$ where $a_{\mathrm{0}}(x)$ , $a_{\mathrm{1}}(x)$ , $p(x)$ , $q(x)$ , $f(x)$ , $g(x)$ , and $h(x)$ are undetermined real functions of $x$ , while $b_{\mathrm{1}}$ is a constant to be determined later. Substituting (13) and (14) into (11) and (12) and collecting all terms with the same order of $t^{\vartheta }{\mathrm{e}}^{j\xi }(\vartheta =\mathrm{0,1},\mathrm{2};j=\mathrm{0,1},\mathrm{2},\cdots )$ together, we derive a set of nonlinear PDEs for $a_{\mathrm{0}}(x)$ , $a_{\mathrm{1}}(x)$ , $p(x)$ , $q(x)$ , $f(x)$ , $g(x)$ , and $h(x)$ , from which we have $$\begin{array}{c}\text{(15)}& a_{\mathrm{0}}=\pm \frac{\mathrm{1}}{\mathrm{2}}q\left(x\right)\sqrt{-\frac{\alpha \left(x\right)}{\beta \left(x\right)}},a_{\mathrm{1}}=\mp \frac{\mathrm{1}}{\mathrm{2}}{b}_{\mathrm{1}}q\left(x\right)\sqrt{-\frac{\alpha \left(x\right)}{\beta \left(x\right)}},\\ \text{(16)}& f\left(x\right)=\frac{\mathrm{1}}{f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x},g\left(x\right)=\frac{g_{\mathrm{0}}}{f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x},\\ \text{(17)}& h\left(x\right)=h_{\mathrm{0}}+\frac{\mathrm{2}{g}_{\mathrm{0}}^{\mathrm{2}}+q_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{8}\left(f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x\right)},\text{(18)}& p\left(x\right)=\frac{g_{\mathrm{0}}{q}_{\mathrm{0}}}{\mathrm{2}\left(f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x\right)},q\left(x\right)=\frac{q_{\mathrm{0}}}{f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x},\end{array}$$ under the constraint $$\begin{array}{}\text{(19)}& \gamma \left(x\right)=-\frac{\alpha \left(x\right)}{f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x}+\frac{{\alpha }^{\prime }\left(x\right)}{\mathrm{2}\alpha \left(x\right)}-\frac{{\beta }^{\prime }\left(x\right)}{\mathrm{2}\beta \left(x\right)},\end{array}$$ where $b_{\mathrm{1}}$ , $f_{\mathrm{0}}$ , $h_{\mathrm{0}}$ , $g_{\mathrm{0}}$ , and $q_{\mathrm{0}}$ are arbitrary constants.

We therefore obtain a pair of rational exp-function solutions of the gNLS in (1): $$\begin{array}{}\text{(20)}& \psi =\pm \frac{g_{\mathrm{0}}{q}_{\mathrm{0}}\left(\mathrm{1}\mp b_{\mathrm{1}}{\mathrm{e}}^{\xi }\right)}{\mathrm{2}\left(f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x\right)\left(\mathrm{1}+b_{\mathrm{1}}{\mathrm{e}}^{\xi }\right)}\sqrt{-\frac{\alpha \left(x\right)}{\beta \left(x\right)}}\exp \left\{i\left[\frac{\mathrm{2}{\left(\mathrm{2}t+g_{\mathrm{0}}\right)}^{\mathrm{2}}+q_{\mathrm{0}}^{\mathrm{2}}}{\mathrm{8}\left(f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x\right)}+h_{\mathrm{0}}\right]\right\},\end{array}$$ where $$\begin{array}{}\text{(21)}& \xi =\frac{q_{\mathrm{0}}}{f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x}+\frac{g_{\mathrm{0}}{q}_{\mathrm{0}}}{\mathrm{2}\left(f_{\mathrm{0}}+\mathrm{2}\int \alpha \left(x\right)\mathrm{d}x\right)}t.\end{array}$$

3.2. N-Wave Solutions

For convenience, we let $$\begin{array}{c}\text{(22)}& X=\frac{\mathrm{1}}{\mathrm{2}}\int \alpha \left(x\right)\mathrm{d}x,T=t,\\ \beta \left(x\right)=\frac{\mathrm{1}}{\mathrm{2}}\alpha \left(x\right){\mathrm{e}}^{-\int \alpha \left(x\right)\mathrm{d}x},\\ \gamma \left(x\right)=\frac{\mathrm{1}}{\mathrm{2}}\alpha \left(x\right),\\ \psi =\lambda e^{\left(\mathrm{1}+\mathrm{i}{\lambda }^{\mathrm{2}}\right)X}\stackrel{-}{\psi },\end{array}$$ and we still write $\stackrel{-}{\psi }$ as $\psi$ ; then (1) is converted into $$\begin{array}{}\text{(23)}& \mathrm{i}{\psi }_X+{\psi }_{TT}+{\lambda }^{\mathrm{2}}\left({\left|\psi \right|}^{\mathrm{2}}-\mathrm{1}\right)\psi =\mathrm{0}.\end{array}$$

In what follows, we construct N-wave solutions of (23). To begin with the single-wave solution, we suppose that $$\begin{array}{c}\text{(24)}& \psi =\frac{\mathrm{1}}{\lambda }{\mathrm{e}}^{-\mathrm{i}{\lambda }^{\mathrm{2}}X}\frac{a_{\mathrm{0}}+a_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}+a_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}^{\ast }}+a_{\mathrm{3}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }}}{\mathrm{1}+b_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}+b_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}^{\ast }}+b_{\mathrm{3}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }}},{\xi }_{\mathrm{1}}=k_{\mathrm{1}}T+c_{\mathrm{1}}X+d_{\mathrm{1}}.\end{array}$$

Substituting (24) into (23) and equating each coefficient of the same order power of ${\mathrm{e}}^{\theta {\xi }_{\mathrm{1}}+\vartheta {\xi }_{\mathrm{1}}^{\ast }}$ ( $\theta ,\vartheta =\mathrm{0,1},\mathrm{2},\cdots$ ) to zero yield a set of algebraic equations for $a_{\mathrm{0}}$ , $a_{\mathrm{1}}$ , $a_{\mathrm{2}}$ , $a_{\mathrm{3}}$ , $b_{\mathrm{1}}$ , $b_{\mathrm{2}}$ , $b_{\mathrm{3}}$ , $c_{\mathrm{1}}$ , and $k_{\mathrm{1}}$ . Solving the set of equations, we have $$\begin{array}{c}\text{(25)}& a_{\mathrm{0}}=\mathrm{0},a_{\mathrm{2}}=\mathrm{0},\\ b_{\mathrm{1}}=\mathrm{0},\\ b_{\mathrm{2}}=\mathrm{0},\\ c_{\mathrm{1}}=\mathrm{i}{k}_{\mathrm{1}}^{\mathrm{2}},\\ b_{\mathrm{3}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{1}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},\end{array}$$ and hence we obtain the single-wave solution: $$\begin{array}{c}\text{(26)}& \psi =\frac{\mathrm{1}}{\lambda }{\mathrm{e}}^{-\mathrm{i}{\lambda }^{\mathrm{2}}X}\frac{a_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}}{\mathrm{1}+a_{\mathrm{1}}{a}_{\mathrm{1}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }+A_{\mathrm{13}}}},{\xi }_{\mathrm{1}}=k_{\mathrm{1}}T+\mathrm{i}{k}_{\mathrm{1}}^{\mathrm{2}}X+d_{\mathrm{1}},\end{array}$$ where $a_{\mathrm{1}}$ , $k_{\mathrm{1}}$ , and $d_{\mathrm{1}}$ are arbitrary complex constants and $$\begin{array}{}\text{(27)}& {\mathrm{e}}^{A_{\mathrm{13}}}=\frac{\mathrm{1}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}}.\end{array}$$

For the double-wave solution, we next suppose that $$\begin{array}{}\text{(28)}& \psi =\frac{\mathrm{1}}{\lambda }{\mathrm{e}}^{-\mathrm{i}{\lambda }^{\mathrm{2}}X}\frac{a_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}+a_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}}+a_{\mathrm{3}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }}+a_{\mathrm{4}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{2}}^{\ast }}}{\mathrm{1}+b_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }}+b_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{3}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }}+b_{\mathrm{4}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{5}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{2}}^{\ast }}},\end{array}$$ where ${\xi }_{\mathrm{1}}=k_{\mathrm{1}}T+c_{\mathrm{1}}X+d_{\mathrm{1}}$ and ${\xi }_{\mathrm{2}}=k_{\mathrm{2}}T+c_{\mathrm{2}}X+d_{\mathrm{2}}$ . Substituting (28) into (23) and equating each coefficient of the same order power of ${\mathrm{e}}^{\theta {\xi }_{\mathrm{1}}+\vartheta {\xi }_{\mathrm{2}}+\mu {\xi }_{\mathrm{1}}^{\ast }+\rho {\xi }_{\mathrm{2}}^{\ast }}(\theta ,\vartheta ,\mu ,\rho =\mathrm{0,1},\mathrm{2},\cdots )$ to zero yield a set of algebraic equations, from which we have $$\begin{array}{c}\text{(29)}& a_{\mathrm{3}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},a_{\mathrm{4}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(30)}& b_{\mathrm{1}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{1}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},b_{\mathrm{2}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ b_{\mathrm{3}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},\\ b_{\mathrm{4}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{2}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(31)}& b_{\mathrm{5}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\text{(32)}& c_{\mathrm{1}}=\mathrm{i}{k}_{\mathrm{1}}^{\mathrm{2}},c_{\mathrm{2}}=\mathrm{i}{k}_{\mathrm{2}}^{\mathrm{2}},\end{array}$$ and hence we obtain the double-wave solution as follows: $$\begin{array}{}\text{(33)}& \psi =\frac{\mathrm{1}}{\lambda }{\mathrm{e}}^{-\mathrm{i}{\lambda }^{\mathrm{2}}X}\frac{F\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast }\right)}{H\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast }\right)},\end{array}$$ where $$\begin{array}{c}\text{(34)}& F\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast }\right)=a_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}+a_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}}+a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }+A_{\mathrm{12}}+A_{\mathrm{13}}+A_{\mathrm{23}}}+a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{2}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{2}}^{\ast }+A_{\mathrm{12}}+A_{\mathrm{14}}+A_{\mathrm{24}}},\text{(35)}& H\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast }\right)=\mathrm{1}+a_{\mathrm{1}}{a}_{\mathrm{1}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }+A_{\mathrm{13}}}+a_{\mathrm{1}}{a}_{\mathrm{2}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}^{\ast }+A_{\mathrm{14}}}+a_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }+A_{\mathrm{23}}}+a_{\mathrm{2}}{a}_{\mathrm{2}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{2}}^{\ast }+A_{\mathrm{24}}}+a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{2}}^{\ast }{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{2}}^{\ast }+A_{\mathrm{12}}+A_{\mathrm{13}}+A_{\mathrm{14}}+A_{\mathrm{23}}+A_{\mathrm{24}}+A_{\mathrm{34}}},\text{(36)}& {\xi }_{\mathrm{1}}=k_{\mathrm{1}}T+\mathrm{i}{k}_{\mathrm{1}}^{\mathrm{2}}X+d_{\mathrm{1}},{\xi }_{\mathrm{2}}=k_{\mathrm{2}}T+\mathrm{i}{k}_{\mathrm{2}}^{\mathrm{2}}X+d_{\mathrm{2}},\\ \text{(37)}& {\mathrm{e}}^{A_{\mathrm{12}}}=\mathrm{2}{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}},{\mathrm{e}}^{A_{j,l+\mathrm{2}}}=\frac{\mathrm{1}}{\mathrm{2}{\left(k_j+k_l^{\ast }\right)}^{\mathrm{2}}},\\ \hspace{1em}\left(j,l=\mathrm{1,2}\right),\\ \text{(38)}& {\mathrm{e}}^{A_{\mathrm{34}}}=\mathrm{2}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}},\end{array}$$ and $a_{\mathrm{1}}$ , $a_{\mathrm{2}}$ , $c_{\mathrm{1}}$ , $c_{\mathrm{2}}$ , $k_{\mathrm{1}}$ , $k_{\mathrm{2}}$ , $d_{\mathrm{1}}$ , and $d_{\mathrm{2}}$ are arbitrary complex constants.

Finally, we determine the three-wave solution of the following form: $$\begin{array}{}\text{(39)}& \psi =\frac{\mathrm{1}}{\lambda }{\mathrm{e}}^{-\mathrm{i}{\lambda }^{\mathrm{2}}X}\frac{F\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{3}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast },{\xi }_{\mathrm{3}}^{\ast }\right)}{H\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{3}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast },{\xi }_{\mathrm{3}}^{\ast }\right)},\end{array}$$ where ${\xi }_{\mathrm{1}}=k_{\mathrm{1}}T+c_{\mathrm{1}}X+d_{\mathrm{1}}$ , ${\xi }_{\mathrm{2}}=k_{\mathrm{2}}T+c_{\mathrm{2}}X+d_{\mathrm{2}}$ , ${\xi }_{\mathrm{3}}=k_{\mathrm{3}}T+c_{\mathrm{3}}X+d_{\mathrm{3}}$ , and $$\begin{array}{}\text{(40)}& F\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{3}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast },{\xi }_{\mathrm{3}}^{\ast }\right)=a_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}}+a_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}}+a_{\mathrm{3}}{\mathrm{e}}^{{\xi }_{\mathrm{3}}}+a_{\mathrm{4}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }}+a_{\mathrm{5}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{2}}^{\ast }}+a_{\mathrm{6}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}^{\ast }}+a_{\mathrm{7}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }}+a_{\mathrm{8}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{2}}^{\ast }}+a_{\mathrm{9}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{3}}^{\ast }}+a_{\mathrm{10}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }}+a_{\mathrm{11}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{2}}^{\ast }}+a_{\mathrm{12}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{3}}^{\ast }}+a_{\mathrm{13}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{2}}^{\ast }}+a_{\mathrm{14}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}+a_{\mathrm{15}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{2}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }},\text{(41)}& H\left({\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},{\xi }_{\mathrm{3}},{\xi }_{\mathrm{1}}^{\ast },{\xi }_{\mathrm{2}}^{\ast },{\xi }_{\mathrm{3}}^{\ast }\right)=\mathrm{1}+b_{\mathrm{1}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{1}}^{\ast }}+b_{\mathrm{2}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{3}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{4}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }}+b_{\mathrm{5}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{6}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{7}}{\mathrm{e}}^{{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }}+b_{\mathrm{8}}{\mathrm{e}}^{{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{9}}{\mathrm{e}}^{{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{10}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{11}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{12}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{2}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{13}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{14}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{15}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{2}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{16}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{2}}^{\ast }}+b_{\mathrm{17}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{18}}{\mathrm{e}}^{{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{2}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}+b_{\mathrm{19}}{\mathrm{e}}^{{\xi }_{\mathrm{1}}+{\xi }_{\mathrm{2}}+{\xi }_{\mathrm{3}}+{\xi }_{\mathrm{1}}^{\ast }+{\xi }_{\mathrm{2}}^{\ast }+{\xi }_{\mathrm{3}}^{\ast }}.\end{array}$$

Similarly, we have $$\begin{array}{c}\text{(42)}& a_{\mathrm{4}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},a_{\mathrm{5}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ a_{\mathrm{6}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(43)}& a_{\mathrm{7}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},a_{\mathrm{8}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{3}}{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ a_{\mathrm{9}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{3}}{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(44)}& a_{\mathrm{10}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},a_{\mathrm{11}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ a_{\mathrm{12}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(45)}& a_{\mathrm{13}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\text{(46)}& a_{\mathrm{14}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(47)}& a_{\mathrm{15}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{2}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(48)}& b_{\mathrm{1}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{1}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},b_{\mathrm{2}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ b_{\mathrm{3}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{3}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(49)}& b_{\mathrm{4}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},b_{\mathrm{5}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{2}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ b_{\mathrm{6}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{3}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(50)}& b_{\mathrm{7}}=\frac{a_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}},b_{\mathrm{8}}=\frac{a_{\mathrm{3}}{a}_{\mathrm{2}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\\ b_{\mathrm{9}}=\frac{a_{\mathrm{3}}{a}_{\mathrm{3}}^{\ast }}{\mathrm{2}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\\ \text{(51)}& b_{\mathrm{10}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\text{(52)}& b_{\mathrm{11}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(53)}& b_{\mathrm{12}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{2}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(54)}& b_{\mathrm{13}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\text{(55)}& b_{\mathrm{14}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(56)}& b_{\mathrm{15}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{3}}{a}_{\mathrm{2}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{1}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(57)}& b_{\mathrm{16}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{2}}^{\ast }{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}},\text{(58)}& b_{\mathrm{17}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(59)}& b_{\mathrm{18}}=\frac{a_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{2}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{4}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(60)}& b_{\mathrm{19}}=\frac{a_{\mathrm{1}}{a}_{\mathrm{2}}{a}_{\mathrm{3}}{a}_{\mathrm{1}}^{\ast }{a}_{\mathrm{2}}^{\ast }{a}_{\mathrm{3}}^{\ast }{\left(k_{\mathrm{1}}-k_{\mathrm{2}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}-k_{\mathrm{3}}\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{1}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}^{\ast }-k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}}{\mathrm{8}{\left(k_{\mathrm{1}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{2}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{1}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{2}}^{\ast }\right)}^{\mathrm{2}}{\left(k_{\mathrm{3}}+k_{\mathrm{3}}^{\ast }\right)}^{\mathrm{2}}},\text{(61)}& c_{\mathrm{1}}=\mathrm{i}{k}_{\mathrm{1}}^{\mathrm{2}},c_{\mathrm{2}}=\mathrm{i}{k}_{\mathrm{2}}^{\mathrm{2}},\\ c_{\mathrm{3}}=\mathrm{i}{k}_{\mathrm{3}}^{\mathrm{2}}.\end{array}$$

With the help of (40)-(61), the three-wave solution (39) can be finally determined as follows: $$\begin{array}{}\text{(62)}& \psi =\frac{\mathrm{1}}{\lambda }{\mathrm{e}}^{-\mathrm{i}{\lambda }^{\mathrm{2}}X}\frac{{\sum }_{\mu =\mathrm{0,1}}{B}_{\mathrm{1}}\left(\mu \right){\prod }_{j=\mathrm{1}}^{\mathrm{6}}{a}_j^{{\mu }_j}{\mathrm{e}}^{{\sum }_{j=\mathrm{1}}^{\mathrm{6}}{\mu }_j{\xi }_j+{\sum }_{j=\mathrm{1}}^{\mathrm{6}}{\mu }_j{\mu }_l{A}_{jl}}}{{\sum }_{\mu =\mathrm{0,1}}{B}_{\mathrm{2}}\left(\mu \right){\prod }_{j=\mathrm{1}}^{\mathrm{6}}{a}_j^{{\mu }_j}{\mathrm{e}}^{{\sum }_{j=\mathrm{1}}^{\mathrm{6}}{\mu }_j{\xi }_j+{\sum }_{j=\mathrm{1}}^{\mathrm{6}}{\mu }_j{\mu }_l{A}_{jl}}},\end{array}$$ where $$\begin{array}{c}\text{(63)}& {\xi }_j=k_{j}T+\mathrm{i}{k}_j^{\mathrm{2}}X+d_j,{\xi }_{N+j}={\xi }_j^{\ast }=k_j^{\ast }T-\mathrm{i}{k}_j^{\ast \mathrm{2}}X+d_j^{\ast },\\ \hspace{1em}\left(j=\mathrm{1,2},\mathrm{3}\right),\\ \text{(64)}& {\mathrm{e}}^{A_{jl}}=\mathrm{2}{\left(k_j-k_l\right)}^{\mathrm{2}},\hspace{1em}\left(j<l=\mathrm{2,3}\right),\text{(65)}& {\mathrm{e}}^{A_{j,N+l}}=\frac{\mathrm{1}}{\mathrm{2}{\left(k_j+k_l^{\ast }\right)}^{\mathrm{2}}},\hspace{1em}\left(j,l=\mathrm{1,2},\mathrm{3}\right),\text{(66)}& {\mathrm{e}}^{A_{N+j,N+l}}={\mathrm{e}}^{A_{j,l}^{\ast }}=\mathrm{2}{\left(k_j^{\ast }-k_l^{\ast }\right)}^{\mathrm{2}},\hspace{1em}\left(j<l=\mathrm{2,3}\right),\text{(67)}& a_{N+j}=a_j^{\ast },\hspace{1em}\left(j=\mathrm{1,2},\cdots ,N\right),\end{array}$$ and the summation ${\Sigma }_{\mu =\mathrm{0,1}}$ refers to all the combinations of ${\mu }_j=\mathrm{0,1}(j=\mathrm{1,2},\mathrm{3})$ ; $B_{\mathrm{1}}(\mu )$ and $B_{\mathrm{2}}(\mu )$ denote that all the following conditions must hold: $$\begin{array}{c}\text{(68)}& {\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{3}}{\mu }_j={\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{3}}{\mu }_{\mathrm{3}+j},{\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{3}}{\mu }_j={\displaystyle \sum }_{j=\mathrm{1}}^{\mathrm{3}}{\mu }_{\mathrm{3}+j}+\mathrm{1}.\end{array}$$

Generally, introducing the notations $$\begin{array}{c}\text{(69)}& {\xi }_j=k_{j}T+\mathrm{i}{k}_j^{\mathrm{2}}X+d_j,{\xi }_{N+j}={\xi }_j^{\ast }=k_j^{\ast }T-\mathrm{i}{k}_j^{\ast \mathrm{2}}X+d_j^{\ast },\\ \hspace{1em}\left(j=\mathrm{1,2},\cdots ,N\right),\\ \text{(70)}& {\mathrm{e}}^{A_{jl}}=\mathrm{2}{\left(k_j-k_l\right)}^{\mathrm{2}},\hspace{1em}\left(j<l=\mathrm{2,3},\cdots ,N\right),\\ \text{(71)}& {\mathrm{e}}^{A_{j,N+l}}=\frac{\mathrm{1}}{\mathrm{2}{\left(k_j+k_l^{\ast }\right)}^{\mathrm{2}}},\hspace{1em}\left(j,l=\mathrm{1,2},\cdots ,N\right),\\ \text{(72)}& {\mathrm{e}}^{A_{N+j,N+l}}={\mathrm{e}}^{A_{j,l}^{\ast }}=\mathrm{2}{\left(k_j^{\ast }-k_l^{\ast }\right)}^{\mathrm{2}},\hspace{1em}\left(j<l=\mathrm{2,3},\cdots ,N\right),\\ \text{(73)}& a_{N+j}=a_j^{\ast },\hspace{1em}\left(j=\mathrm{1,2},\cdots ,N\right),\end{array}$$ we can obtain a uniform formula of the N-wave solution: $$\begin{array}{}\text{(74)}& \psi =\frac{\mathrm{1}}{\lambda }{\mathrm{e}}^{-\mathrm{i}{\lambda }^{\mathrm{2}}X}\frac{{\sum }_{\mu =\mathrm{0,1}}{B}_{\mathrm{1}}\left(\mu \right){\prod }_{j=\mathrm{1}}^{\mathrm{2}N}{a}_j^{{\mu }_j}{\mathrm{e}}^{{\sum }_{j=\mathrm{1}}^{\mathrm{2}N}{\mu }_j{\xi }_j+{\sum }_{j=\mathrm{1}}^{\mathrm{2}N}{\mu }_j{\mu }_l{A}_{jl}}}{{\sum }_{\mu =\mathrm{0,1}}{B}_{\mathrm{2}}\left(\mu \right){\prod }_{j=\mathrm{1}}^{\mathrm{2}N}{a}_j^{{\mu }_j}{\mathrm{e}}^{{\sum }_{j=\mathrm{1}}^{\mathrm{2}N}{\mu }_j{\xi }_j+{\sum }_{j=\mathrm{1}}^{\mathrm{2}N}{\mu }_j{\mu }_l{A}_{jl}}},\end{array}$$ where the summation ${\Sigma }_{\mu =\mathrm{0,1}}$ refers to all the combinations of ${\mu }_j=\mathrm{0,1}(j=\mathrm{1,2},\cdots ,N)$ ; $B_{\mathrm{1}}(\mu )$ and $B_{\mathrm{2}}(\mu )$ denote that all the following conditions must hold: $$\begin{array}{c}\text{(75)}& {\displaystyle \sum }_{j=\mathrm{1}}^N{\mu }_j={\displaystyle \sum }_{j=\mathrm{1}}^N{\mu }_{N+j},{\displaystyle \sum }_{j=\mathrm{1}}^N{\mu }_j={\displaystyle \sum }_{j=\mathrm{1}}^N{\mu }_{N+j}+\mathrm{1}.\end{array}$$

3.3. Rouge Wave Solutions

To construct rouge wave solutions, we rewrite (9) as $$\begin{array}{}\text{(76)}& \psi =\frac{a_{\mathrm{0}}+a_{\mathrm{1}}\cos kT{\mathrm{e}}^{cX+d}+a_{\mathrm{2}}{\mathrm{e}}^{\mathrm{2}cX+\mathrm{2}d}}{\mathrm{1}+b_{\mathrm{1}}\cos kT{\mathrm{e}}^{cX+d}+b_{\mathrm{2}}{\mathrm{e}}^{\mathrm{2}cX+\mathrm{2}d}},\end{array}$$ and we substitute (76) into (23); then we equate each coefficient of the same order power of $\mathrm{c}\mathrm{o}{\mathrm{s}}^{\mu }kT{\mathrm{e}}^{\theta (cX+d)}(\mu ,\theta =\mathrm{0,1},\mathrm{2},\cdots )$ to zero; a set of algebraic equations is derived. Solving the set of equations, we have $$\begin{array}{c}\text{(77)}& a_{\mathrm{0}}=\mathrm{1},a_{\mathrm{1}}=\frac{\lambda \sqrt{\mathrm{2}{b}_{\mathrm{2}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}}{{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}},\\ a_{\mathrm{2}}=\frac{b_{\mathrm{2}}\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\mp \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)}{{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}},\\ \text{(78)}& b_{\mathrm{1}}=\frac{\sqrt{\mathrm{2}{b}_{\mathrm{2}}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}{\lambda },c=\pm k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}},\end{array}$$ and $$\begin{array}{c}\text{(79)}& a_{\mathrm{0}}=\mathrm{1},a_{\mathrm{1}}=-\frac{\lambda \sqrt{\mathrm{2}{b}_{\mathrm{2}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}}{{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}},\\ a_{\mathrm{2}}=\frac{b_{\mathrm{2}}\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\mp \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)}{{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}},\\ \text{(80)}& b_{\mathrm{1}}=-\frac{\sqrt{\mathrm{2}{b}_{\mathrm{2}}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}{\lambda },c=\pm k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}.\end{array}$$

We, therefore, obtain two pairs of rational exp-function wave solutions as follows: $$\begin{array}{}\text{(81)}& \psi =\frac{\mathrm{1}+\left(\lambda \sqrt{\mathrm{2}{b}_{\mathrm{2}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}/\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)\right)\cos kT{\mathrm{e}}^{\pm k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+d}+\left(b_{\mathrm{2}}\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\mp \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)/\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)\right){\mathrm{e}}^{\pm \mathrm{2}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+\mathrm{2}d}}{\mathrm{1}+\left(\sqrt{\mathrm{2}{b}_{\mathrm{2}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}/\lambda \right)\cos kT{\mathrm{e}}^{\pm k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+d}+b_{\mathrm{2}}{\mathrm{e}}^{\pm \mathrm{2}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+\mathrm{2}d}},\text{(82)}& \psi =\frac{\mathrm{1}-\left(\lambda \sqrt{\mathrm{2}{b}_{\mathrm{2}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}/\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)\right)\cos kT{\mathrm{e}}^{\pm k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+d}+\left(b_{\mathrm{2}}\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\mp \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)/\left({\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\pm \mathrm{i}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}\right)\right){\mathrm{e}}^{\pm \mathrm{2}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+\mathrm{2}d}}{\mathrm{1}-\left(\sqrt{\mathrm{2}{b}_{\mathrm{2}}\left(\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}\right)}/\lambda \right)\cos kT{\mathrm{e}}^{\pm k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+d}+b_{\mathrm{2}}{\mathrm{e}}^{\pm \mathrm{2}k\sqrt{\mathrm{2}{\lambda }^{\mathrm{2}}-k^{\mathrm{2}}}X+\mathrm{2}d}},\end{array}$$

It is easy to see that when $b_{\mathrm{2}}=\mathrm{1}$ , $d=\mathrm{0}$ , and $\lambda <\mathrm{0}$ the molecular and denominator of solution (81) tend to zeros, respectively. We differentiate (81) with respect to $k$ twice and let $k\to \mathrm{0}$ ; then the limits of solution (81) give two rouge wave solutions: $$\begin{array}{}\text{(83)}& \psi =\frac{\mathrm{9}-\mathrm{8}\mathrm{i}{\lambda }^{\mathrm{2}}t-\mathrm{12}{\lambda }^{\mathrm{4}}{t}^{\mathrm{2}}+\mathrm{2}{\lambda }^{\mathrm{2}}{x}^{\mathrm{2}}}{\mathrm{1}+\mathrm{4}{\lambda }^{\mathrm{4}}{t}^{\mathrm{2}}+\mathrm{2}{\lambda }^{\mathrm{2}}{x}^{\mathrm{2}}},\text{(84)}& \psi =\frac{-\mathrm{3}-\mathrm{24}\mathrm{i}{\lambda }^{\mathrm{2}}t+\mathrm{4}{\lambda }^{\mathrm{4}}{t}^{\mathrm{2}}+\mathrm{2}{\lambda }^{\mathrm{2}}{x}^{\mathrm{2}}}{\mathrm{1}+\mathrm{4}{\lambda }^{\mathrm{4}}{t}^{\mathrm{2}}+\mathrm{2}{\lambda }^{\mathrm{2}}{x}^{\mathrm{2}}}.\end{array}$$

In a similar way, when $b_{\mathrm{2}}=\mathrm{1}$ , $d=\mathrm{0}$ , and $\lambda >\mathrm{0}$ , the limits of solution (82) give two rouge wave solutions, which are the same as those in (83) and (84), respectively.

4. 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 $b_{\mathrm{1}}=\mathrm{1}$ , $f_{\mathrm{0}}=\mathrm{1}$ , $g_{\mathrm{0}}=\mathrm{0.1}$ , $h_{\mathrm{0}}=\mathrm{2}$ , $\alpha (x)=x^{\mathrm{3}}$ , and $\beta (x)=-x$ ; then the modules of solution (20) with “+” branch and different values of $q_{\mathrm{0}}$ 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 $q_{\mathrm{0}}$ is, the smaller the influence on M-shape wave will be.

[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Secondly, we consider solutions (26), (33), and (62). In Figure 5, the module of the single-wave solution (26) is shown by selecting $a_{\mathrm{1}}=\mathrm{1}+\mathrm{i}$ , $k_{\mathrm{1}}=\mathrm{1}-\mathrm{0.2}\mathrm{i}$ , and $\lambda =\mathrm{1}$ . We simulate the module of the double-wave solution (33) in Figure 6, where $a_{\mathrm{1}}=\mathrm{0.5}$ , $a_{\mathrm{2}}=\mathrm{1}$ , $k_{\mathrm{1}}=\mathrm{1}$ , $k_{\mathrm{2}}=\mathrm{0.2}$ , and $\lambda =\mathrm{1}$ . Selecting $a_{\mathrm{1}}=\mathrm{1}$ , $a_{\mathrm{2}}=\mathrm{0.5}$ , $a_{\mathrm{3}}=\mathrm{1.5}$ , $k_{\mathrm{1}}=\mathrm{1}$ , $k_{\mathrm{2}}=\mathrm{0.5}$ , $k_{\mathrm{3}}=-\mathrm{0.25}$ , and $\lambda =\mathrm{1}$ , we show the module of the three-wave solution (62) in Figure 7.

[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Thirdly, in Figures 8–17, we simulate some modules of one branch of solution (81) by selecting $\lambda =\mathrm{1}$ and different values of $k$ . We can see from Figures 8–17 that the value of $k$ has influenced the spatial structures of the module of solution (81) and can also lead to the periodicity and singularity.

[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Finally, we simulate the rouge wave solutions (83) and (84). In Figure 18, a rouge wave structure determined by the module of solution (83) is shown by selecting $\lambda =-\mathrm{1}$ . We show the contour of the rouge wave structure and dynamical evolutions determined by the module of solution (83) in Figures 19 and 20, respectively. At the same time, in Figure 21, we show another rouge wave structure determined by the module of solution (84) by selecting $\lambda =-\mathrm{1}$ . In Figures 22 and 23, the contour of the rouge wave structure and dynamical evolutions determined by the module of solution (84) are shown.

[figure omitted; refer to PDF] [figure omitted; refer to PDF]

[figures omitted; refer to PDF]

[figure omitted; refer to PDF] [figure omitted; refer to PDF]

[figures omitted; refer to PDF]

5. Conclusion

In summary, we have obtained explicit solitary wave solutions, N-wave solutions, and rouge wave solutions of the gNLS in (1) benefiting from the complex multirational exp-function ansatz (4) presented in this paper for nonlinear PDEs with complex coefficients. To the best of our knowledge, the exp-function method and its improvements [8–11] have not been used for the gNLS in (1) and the obtained solutions with free parameters have not been reported in the literatures. In 2001, Serkin and Belyaeva derived a new and more general NLS equation [25]: $$\begin{array}{}\text{(85)}& \mathrm{i}{q}_t=\frac{\mathrm{1}}{\mathrm{2}}D\left(t\right)q_{xx}+\beta R\left(t\right){\left|q\right|}^{\mathrm{2}}q-\mathrm{2}\alpha \left(t\right)xq+\frac{\mathrm{i}}{\mathrm{2}}\frac{W\left[R\left(t\right),D\left(t\right)\right]}{R\left(t\right)D\left(t\right)}q-\mathrm{i}V\left(t\right)q_t-\mathrm{2}\Gamma \left(t\right)q,\end{array}$$ as the condition for integrability of a pair of linear differential equations; here the Wronskian $W[R(t),D(t)]=R(t)\mathrm{d}D(t)/\mathrm{d}t-D(t)\mathrm{d}R(t)/\mathrm{d}t$ . Since (85) is Lax-integrable [25] and the gNLS in (1) can be reduced from (85), the Lax-integrability of the gNLS in (1) is obvious. As for the NLS equations with varying coefficients, their Lax-representations, and other investigations, we can refer to a lot of references such as those in [25–38]. Compared with the inverse scattering method [1], Hirota’s bilinear method [2], and Darboux transformation [3], the complex multirational exp-function ansatz (4) for constructing solitary wave solutions, N-wave solutions, and rouge wave solutions does not carry out the processes of bilinearization and spectral problems. In spite of this, the approach needs a lot of calculations and even has a lot of uncertainties. Recently, nonlinear PDEs with fractional derivatives and their applications have attracted much attention [39–53]. How to extend the complex multirational exp-function ansatz (4) to such fractional PDEs and the NLS equations with varying coefficients in [25–38] is worthy of study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11547005), the Natural Science Foundation of Liaoning Province of China (20170540007), the Natural Science Foundation of Education Department of Liaoning Province of China (LZ2017002), and Innovative Talents Support Program in Colleges and Universities of Liaoning Province (LR2016021).

References

[1] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, "Method for solving the Korteweg-deVries equation," Physical Review Letters, vol. 19 no. 19, pp. 1095-1097, DOI: 10.1103/PhysRevLett.19.1095, 1967.

[2] R. Hirota, "Exact solution of the korteweg—de vries equation for multiple Collisions of solitons," Physical Review Letters, vol. 27 no. 18, pp. 1192-1194, DOI: 10.1103/PhysRevLett.27.1192, 1971.

[3] S. Zhang, D.-D. Liu, "The third kind of Darboux transformation and multisoliton solutions for generalized Broer-Kaup equations," Turkish Journal of Physics, vol. 39 no. 2, pp. 165-177, DOI: 10.3906/fiz-1411-9, 2015.

[4] M. Wang, "Exact solutions for a compound KdV-Burgers equation," Physics Letters A, vol. 213 no. 5-6, pp. 279-287, DOI: 10.1016/0375-9601(96)00103-X, 1996.

[5] E. G. Fan, "Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems," Physics Letters A, vol. 300 no. 2-3, pp. 243-249, DOI: 10.1016/s0375-9601(02)00776-4, 2002.

[6] E. G. Fan, "Extended tanh-function method and its applications to nonlinear equations," Physics Letters A, vol. 277 no. 4-5, pp. 212-218, DOI: 10.1016/S0375-9601(00)00725-8, 2000.

[7] Z. Yan, H. Zhang, "New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water," Physics Letters A, vol. 285 no. 5-6, pp. 355-362, DOI: 10.1016/S0375-9601(01)00376-0, 2001.

[8] J. He, X. Wu, "Exp-function method for nonlinear wave equations," Chaos, Solitons & Fractals, vol. 30 no. 3, pp. 700-708, DOI: 10.1016/j.chaos.2006.03.020, 2006.

[9] S. Zhang, Z. Y. Wang, "Exact solutions of the nonlinear Schrödinger equation by generalizing exp-function method," Proceedings of the 2015 International Conference on Applied Mathematics and Computational Methods in Engineering (AMCME’15), pp. 74-82, .

[10] I. Aslan, "Multi-wave and rational solutions for nonlinear evolution equations," International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11 no. 8, pp. 619-623, 2010.

[11] I. Aslan, "Rational and multi-wave solutions to some nonlinear physical models," Romanian Journal of Physics, vol. 58 no. 7-8, pp. 893-903, 2013.

[12] S. Zhang, T. Xia, "A generalized F -expansion method and new exact solutions of Konopelchenko-Dubrovsky equations," Applied Mathematics and Computation, vol. 183 no. 2, pp. 1190-1200, DOI: 10.1016/j.amc.2006.06.043, 2006.

[13] C.-Q. Dai, Y.-Y. Wang, "Controllable combined Peregrine soliton and Kuznetsov-Ma soliton in PT-symmetric nonlinear couplers with gain and loss," Nonlinear Dynamics, vol. 80 no. 1-2, pp. 715-721, DOI: 10.1007/s11071-015-1900-0, 2015.

[14] S. Zhang, X. D. Gao, "Exact N-soliton solutions and dynamics of a new AKNS equations with time-dependent coefficients," Nonlinear Dynamics, vol. 83 no. 1-2, pp. 1043-1052, DOI: 10.1007/s11071-015-2386-5, 2016.

[15] Sheng Zhang, Siyu Hong, "Lax Integrability and Soliton Solutions for a Nonisospectral Integrodifferential System," Complexity, vol. 2017,DOI: 10.1155/2017/9457078, 2017.

[16] S. Zhang, B. Xu, H.-Q. Zhang, "Exact solutions of a KdV equation hierarchy with variable coefficients," International Journal of Computer Mathematics, vol. 91 no. 7, pp. 1601-1616, DOI: 10.1080/00207160.2013.855730, 2014.

[17] S. Zhang, X. Gao, "Exact solutions and dynamics of a generalized AKNS equations associated with the nonisospectral depending on exponential function," Journal of Nonlinear Sciences and Applications, vol. 9 no. 6, pp. 4529-4541, DOI: 10.22436/jnsa.009.06.91, 2016.

[18] Z. Z. Kang, T. C. Xia, "Multi-solitons for the coupled FokasLenells system via riemannHilbert approach," Chinese Physics Letters, vol. 35 no. 7, 2018.

[19] B. Xu, S. Zhang, "A novel approach to a time-dependent-coefficient wbk system: doubly periodic waves and singular nonlinear dynamics," Complexity, vol. 2018,DOI: 10.1155/2018/3158126, 2018.

[20] V. I. Kruglov, A. C. Peacock, J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Physical Review Letters, vol. 90 no. 11,DOI: 10.1103/PhysRevE.71.056619, 2003.

[21] S. Zhang, T.-C. Xia, "A generalized auxiliary equation method and its application to (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Vesselov equations," Journal of Physics A: Mathematical and General, vol. 40 no. 2, pp. 227-248, DOI: 10.1088/1751-8113/40/2/003, 2007.

[22] V. N. Serkin, A. Hasegawa, "Novel soliton solutions of the nonlinear Schrodinger equation model," Physical Review Letters, vol. 85 no. 21, pp. 4502-4505, DOI: 10.1103/PhysRevLett.85.4502, 2000.

[23] V. N. Serkin, T. L. Belyaeva, "Optimal control of dark solitons," Optik - International Journal for Light and Electron Optics, vol. 168, pp. 827-838, DOI: 10.1016/j.ijleo.2018.05.003, 2018.

[24] V. I. Kruglov, J. D. Harvey, "Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters," Journal of the Optical Society of America B, vol. 23 no. 12, pp. 2541-2550, DOI: 10.1364/JOSAB.23.002541, 2006.

[25] V. N. Serkin, T. L. Belyaeva, "High-energy optical Schrödinger solitons," JETP Letters, vol. 74 no. 12, pp. 573-577, DOI: 10.1134/1.1455063, 2001.

[26] V. N. Serkin, T. L. Belyaeva, "Optimal control of optical soliton parameters: Part 1. The Lax representation in the problem of soliton management," Quantum Electronics, vol. 31 no. 11, pp. 1007-1015, DOI: 10.1070/QE2001v031n11ABEH002093, 2001.

[27] V. N. Serkin, A. Hasegawa, "Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements," IEEE Journal of Selected Topics in Quantum Electronics, vol. 8 no. 3, pp. 418-431, DOI: 10.1109/JSTQE.2002.1016344, 2002.

[28] V. N. Serkin, A. Hasegawa, T. L. Belyaeva, "Nonautonomous solitons in external potentials," Physical Review Letters, vol. 98 no. 7,DOI: 10.1103/PhysRevLett.98.074102, 2007.

[29] S. K. Turitsyn, E. G. Shapiro, "Dispersion-managed soliton in fiber links with in-line filtering presented in the basis of chirped Gauss-Hermite functions," Journal of the Optical Society of America B: Optical Physics, vol. 16 no. 9, pp. 1321-1331, DOI: 10.1364/JOSAB.16.001321, 1999.

[30] F. Calogero, W. Eckhaus, "Necessary conditions for integrability of nonlinear PDEs," Inverse Problems, vol. 3 no. 2, pp. L27-L32, DOI: 10.1088/0266-5611/3/2/001, 1987.

[31] C. Q. Dai, S. Q. Zhu, L. L. Wang, J. F. Zhang, "Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients," Europhysics Letters, vol. 92 no. 2, 2010.

[32] C.-Q. Dai, Y.-Y. Wang, Q. Tian, J.-F. Zhang, "The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schröinger equation," Annals of Physics, vol. 327 no. 2, pp. 512-521, DOI: 10.1016/j.aop.2011.11.016, 2012.

[33] H.-J. Jiang, J.-J. Xiang, C.-Q. Dai, Y.-Y. Wang, "Nonautonomous bright soliton solutions on continuous wave and cnoidal wave backgrounds in blood vessels," Nonlinear Dynamics, vol. 75 no. 1-2, pp. 201-207, DOI: 10.1007/s11071-013-1058-6, 2014.

[34] T. I. Lakoba, D. J. Kaup, "Hermite-gaussian expansion for pulse propagation in strongly dispersion managed fibers," Physical Review E, vol. 58 no. 5, pp. 6728-6741, 1998.

[35] J. C. Liang, H. P. Liu, F. Liu, L. Yi, "Analytical solutions to the (3+1)-dimensional generalized nonlinear Schrödinger equation with varying parameters," Journal of Physics A: Mathematical and Theoretical, vol. 42 no. 33,DOI: 10.1088/1751-8113/42/33/335204, 2009.

[36] A. Kundu, "Integrable nonautonomous nonlinear Schrödinger equations are equivalent to the standard autonomous equation," Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 79 no. 1,DOI: 10.1103/PhysRevE.79.015601, 2009.

[37] Z. Y. Yan, V. V. Konotop, "Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities," Physical Review E, vol. 80 no. 3, 2009.

[38] Z. Yan, V. V. Konotop, N. Akhmediev, "Three-dimensional rogue waves in nonstationary parabolic potentials," Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 82 no. 3,DOI: 10.1103/PhysRevE.82.036610, 2010.

[39] Y. Wang, L. Liu, X. Zhang, Y. Wu, "Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection," Applied Mathematics and Computation, vol. 258, pp. 312-324, DOI: 10.1016/j.amc.2015.01.080, 2015.

[40] Y. Wang, J. Jiang, "Existence and nonexistence of positive solutions for the fractional coupled system involving generalized p-Laplacian," Advances in Difference Equations,DOI: 10.1186/s13662-017-1385-x, 2017.

[41] K. M. Zhang, "On a sign-changing solution for some fractional differential equations," Boundary Value Problems, vol. 2017 no. 59, 2017.

[42] Xinsheng Du, Anmin Mao, "Existence and Multiplicity of Nontrivial Solutions for a Class of Semilinear Fractional Schrödinger Equations," Journal of Function Spaces, vol. 2017,DOI: 10.1155/2017/3793872, 2017.

[43] Y. Q. Wang, L. S. Liu, "Positive solutions for a class of fractional 3-point boundary value problems at resonance," Advances in Difference Equations, vol. 2017, 2017.

[44] X. Zhang, L. Liu, Y. Wu, B. Wiwatanapataphee, "Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion," Applied Mathematics Letters, vol. 66,DOI: 10.1016/j.aml.2016.10.015, 2017.

[45] L. L. Liu, X. Zhang, L. Liu, Y. Wu, "Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions," Advances in Difference Equations,DOI: 10.1186/s13662-016-0876-5, 2016.

[46] L. M. Guo, L. S. Liu, Y. H. Wu, "Existence of positive solutions for singular fractional differential equations with infinite-point boundary conditions," Nonlinear Analysis: Modelling and Control, vol. 21 no. 5, pp. 635-650, 2016.

[47] J. Wu, X. Zhang, L. Liu, Y. Wu, "Twin iterative solutions for a fractional differential turbulent flow model," Boundary Value Problems,DOI: 10.1186/s13661-016-0604-9, 2016.

[48] J. Wu, X. Zhang, L. Liu, Y. Wu, B. Wiwatanapataphee, "Iterative algorithm and estimation of solution for a fractional order differential equation," Boundary Value Problems, vol. 2016 no. 1,DOI: 10.1186/s13661-016-0608-5, 2016.

[49] B. Zhu, L. Liu, Y. Wu, "Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay," Applied Mathematics Letters, vol. 61, pp. 73-79, DOI: 10.1016/j.aml.2016.05.010, 2016.

[50] H. Liu, F. Meng, "Some new generalized Volterra-Fredholm type discrete fractional sum inequalities and their applications," Journal of Inequalities and Applications, vol. 2016 no. 1, article no. 213,DOI: 10.1186/s13660-016-1152-7, 2016.

[51] R. Xu, F. Meng, "Some new weakly singular integral inequalities and their applications to fractional differential equations," Journal of Inequalities and Applications, vol. 2016 no. 78,DOI: 10.1186/s13660-016-1015-2, 2016.

[52] S. Zhang, H. Q. Zhang, "Fractional sub-equation method and its applications to nonlinear fractional PDEs," Physics Letters A, vol. 375 no. 7, pp. 1069-1073, DOI: 10.1016/j.physleta.2011.01.029, 2011.

[53] J.-H. He, "Fractal calculus and its geometrical explanation," Results in Physics, vol. 10, pp. 272-276, DOI: 10.1016/j.rinp.2018.06.011, 2018.