The Multi-Soliton Solutions for the (2+1)-Dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

Abstract

The (2+1)-dimensional integrable Caudrey–Dodd–Gibbon–Kotera–Sawada equation is a higher-order generalization of the Kadomtsev–Petviashvili equation, which can be applied in some physical branches such as the nonlinear dispersive phenomenon. In this paper, we first present the bilinear form for this equation after constructing one Bäcklund transformation. As a result, the one-soliton solution, two-soliton solution, and three-soliton solution are shown successively and the corresponding soliton structures are constructed. These solitons and their interactions illustrate that the obtained solutions have powerful applications.

1. Introduction

Konpelchenko and Dubrovsky first proposed the following (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada (CDGKS) equation:

$$36u_t+{(u_{4x}+15u{u}_{xx}+15u^3)}_x-5(u_{xxy}+3u{u}_y+3u_x{v}_y+v_{yy}),u=v_x,$$

(1)

through the inverse scattering transform method, in which $u=u(x,y,t),v=v(x,y,t)$ [1]. Cheng proved that this (2+1)-dimensional equation can be constrained to the integrable hierarchies in 1+1 dimensions [2]. Hu proposed a hierarchy of the bilinear CDGKS equations with a unified structure and proved one nonlinear superposition formula under certain conditions [3]. For the CDGKS Equation (1), Cao obtained the quasi-periodic solutions in terms of the Riemann theta functions [4,5]. Tang illustrated the lump solitons and their interaction phenomena through a direct method [6], while Peng studied the characteristics of the solitary waves and lump waves with interaction phenomena by the way of vector notations [7]. Fang demonstrated the lump-type solution, rogue wave, fusion and fission phenomena by means of the Hirota direct method, and symbolic computation [8]. Zhang constructed the generalized lump solution, classical lump solution, and the novel analytical solution based on the bilinear neural network method [9]. Ma and Zhuang presented the specific expression of the N-soliton solutions for the (2+1)-dimensional generalized CDGKS equation in fluid mechanics under Hirota’s bilinear method and studied the different localized waves [10,11,12]. The Nth-order Pfaffian solutions via the modified Pfaffian technique, the symmetry group theorem by using the improved Clarkson, and the Kruskal (CK) method were also applied to the above equation [13,14]. The Kadomtsev–Petviashvili (KP) equation corresponds to a third-order Korteweg-de Vries (KdV) integrable equation, while the CDGKS equation is the fifth-order modified KdV integrable equation, which is a higher-order representation of the KP equation [2,15].

We know that to find the solitary waves and solitons for an integrable system, apart from the above-mentioned methods, such as the Hirota bilinear method, the Bäcklund transformation, the direct CK’s method, and the bilinear neural network method, there are still some powerful tools that can be listed, including the inverse scattering method, the Darboux transformation, the Painlevé analysis approach, and the bilinear residual network method [16,17,18,19,20,21,22].

The paper is outlined as follows: In Section 2, the bilinear form is written through a Bäcklund transformation for Equation (1). Therefore, the solutions which contain multi-solitons exist for this equation. The abundant solitons, such as the peak, plateau, M-type, and molecular solitons for the one-soliton solution, the interactions of two peak solitons, two M-type solitons, and two molecular solitons, as well as breathers with their molecules for the two-soliton solution, the interactions of three peak solitons, three molecular solitons, one peak soliton, and a breather for the three-soliton solution are all constructed successively after the constrained parameters are taken suitably. Section 3 is a short summary.

2. The Multi-Soliton Solutions and Their Hybrid Structures

We introduce the following Bäcklund transformation of Equation (1):

$$u=2{(lnf)}_{xx},v=2{(lnf)}_x,$$

(2)

with the transformation function $f=f(x,y,t)$ being defined later. Therefore, Equation (1) has the bilinear form

$$\begin{array}{c}\hfill (36D_x{D}_t+D_x^6-5D_x^3{D}_y-5D_y^2)(f·f)=0,\end{array}$$

(3)

where the Hirota derivative operators are as follows [23,24]:

$$\begin{array}{c}\hfill D_x^U{D}_y^V{D}_t^W(P·Q)={({\displaystyle \frac{\partial }{\partial x}}-{\displaystyle \frac{\partial }{\partial x^{\prime }}})}^U{({\displaystyle \frac{\partial }{\partial y}}-{\displaystyle \frac{\partial }{\partial y^{\prime }}})}^V{({\displaystyle \frac{\partial }{\partial t}}-{\displaystyle \frac{\partial }{\partial t^{\prime }}})}^{W}P(x,y,t)·Q(x^{\prime },y^{\prime },t^{\prime }){|}_{x^{\prime }=x,y=y^{\prime },t^{\prime }=t}.\end{array}$$

(4)

In a similar situation, Equation (3) has another bilinear version as follows:

$$36f_x{f}_t-36f_{xt}f-f_{6x}f+6f_{5x}{f}_x-15f_{4x}{f}_{xx}$$

$$+10f_{3x}^2-5f_{3x}{f}_y+5f_{3xy}f+15f_{xx}{f}_{xy}-15f_{xxy}{f}_x+5f_{yy}f-5f_y^2=0.$$

(5)

2.1. The One-Soliton Solution

Equation (1) has a single soliton solution through Equation (2) with

$$\begin{array}{c}\hfill u={\displaystyle \frac{2k_1^2(b_1{\mathbf{e}}^{{\xi }_1}+4{\delta }_1{\mathbf{e}}^{2{\xi }_1}+b_1{\delta }_1{\mathbf{e}}^{3{\xi }_1})}{f^2}},v={\displaystyle \frac{2k_1(b_1{\mathbf{e}}^{{\xi }_1}+2{\delta }_1{\mathbf{e}}^{2{\xi }_1})}{f}},\end{array}$$

(6)

where

$$\begin{array}{c}\hfill f\equiv f_1=1+b_1{\mathbf{e}}^{{\xi }_1}+{\delta }_1{\mathbf{e}}^{2{\xi }_1},{\xi }_1=k_{1}x+k_1^{3}y+{\displaystyle \frac{1}{4}}{k}_1^{5}t,\end{array}$$

(7)

Here, $k_1,b_1$ and ${\delta }_1$ are three constants. In fact, the parameter ${\delta }_1$ affects the structure of this soliton through its determined function f. For example, the single peak soliton can be presented when ${\delta }_1={\displaystyle \frac{1}{4}}$. As the value of ${\delta }_1$ decreases, the shape of the soliton is changed. Figure 1 and Figure 2 show the evolution process with ${\delta }_1={\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{16}},{\displaystyle \frac{1}{50}}$, and ${\displaystyle \frac{1}{15,000}}$, respectively, with other parameters taking the same value $k_1=b_1=1$ when $t=2$. As a result, the peak soliton (Figure 1a), plateau soliton (Figure 1b), M-type soliton (Figure 2a), and molecular soliton (Figure 2b) will be constructed sooner or later.

2.2. The Two-Soliton Solution

The two-soliton solution of Equation (1) can be expressed through Equation (2):

$$f\equiv f_2=1+b_1{\mathbf{e}}^{{\xi }_1}+b_2{\mathbf{e}}^{{\xi }_2}+{\delta }_1{\mathbf{e}}^{2{\xi }_1}+{\delta }_2{\mathbf{e}}^{2{\xi }_2}+B_{12}{\mathbf{e}}^{{\xi }_1+{\xi }_2}$$

$$\begin{array}{c}\hfill +b_1{\delta }_2{a}_{12}^2{\mathbf{e}}^{{\xi }_1+2{\xi }_2}+b_2{\delta }_1{a}_{12}^2{\mathbf{e}}^{2{\xi }_1+{\xi }_2}+Y_{12}{\mathbf{e}}^{2{\xi }_1+2{\xi }_2},{\xi }_1=k_{1}x+k_1^{3}y+{\displaystyle \frac{1}{4}}{k}_1^{5}t,{\xi }_2=k_{2}x+k_2^{3}y+{\displaystyle \frac{1}{4}}{k}_2^{5}t,\end{array}$$

(8)

where the constraint constants have

$$\begin{array}{c}\hfill a_{12}={\displaystyle \frac{k_1-k_2}{k_1+k_2}},Y_{12}={\delta }_1{\delta }_2{a}_{12}^4,\end{array}$$

(9)

from Equation (5).

For this circumstance, to study the soliton structures for the CDGKS Equation (1) through the auxiliary function of Equation (8), the asymptotic analysis technique should be adopted [25,26,27,28]. That is, when taking the independent parameters $B_{12}=b_1=b_2=k_1=1$ and $k_2={\displaystyle \frac{4}{5}}$, the asymptotic expressions have the following forms:

(1) Before the action $(t\to -\infty )$,

(a) ${\xi }_1\sim 0,{\xi }_2\sim +\infty ,f\sim b_2{\mathbf{e}}^{{\xi }_2}+{\delta }_2{\mathbf{e}}^{2{\xi }_2}+B_{12}{\mathbf{e}}^{{\xi }_1+{\xi }_2}+{\delta }_1{b}_2{a}_{12}^2{\mathbf{e}}^{2{\xi }_1+{\xi }_2}+{\delta }_2{b}_1{a}_{12}^2{\mathbf{e}}^{{\xi }_1+2{\xi }_2}+Y_{12}{\mathbf{e}}^{2{\xi }_1+2{\xi }_2},$

$$u\to u_1^{-}=2(1076168025{\mathbf{e}}^{{\xi }_{10}+2{\xi }_{20}}+656100{\delta }_1{\mathbf{e}}^{2{\xi }_{10}+2{\xi }_{20}}+164025{\delta }_1{\mathbf{e}}^{3{\xi }_{10}+2{\xi }_{20}}+688747536{\delta }_2{\mathbf{e}}^{3{\xi }_{20}}$$

$$+43578162{\delta }_2{\mathbf{e}}^{{\xi }_{10}+3{\xi }_{20}}+104976{\delta }_2{\mathbf{e}}^{2{\xi }_{10}+3{\xi }_{20}}+1522152{\delta }_1{\delta }_2{\mathbf{e}}^{2{\xi }_{10}+3{\xi }_{20}}+531442{\delta }_1{\delta }_2{\mathbf{e}}^{3{\xi }_{10}+3{\xi }_{20}}$$

$$+164025{\delta }_2^2{\mathbf{e}}^{{\xi }_{10}+4{\xi }_{20}}+16{\delta }_1^2{\delta }_2{\mathbf{e}}^{4{\xi }_{10}+3{\xi }_{20}}+656100{\delta }_1{\delta }_2^2{\mathbf{e}}^{2{\xi }_{10}+4{\xi }_{20}}+25{\delta }_1{\delta }_2^2{\mathbf{e}}^{3{\xi }_{10}+4{\xi }_{20}})/25(6561{\mathbf{e}}^{{\xi }_{20}}$$

$$\begin{array}{c}\hfill +6561{\mathbf{e}}^{{\xi }_{10}+{\xi }_{20}}+{\delta }_1{\mathbf{e}}^{2{\xi }_{10}+{\xi }_{20}}+{\delta }_2{\mathbf{e}}^{{\xi }_{10}+2{\xi }_{20}}+6561{\delta }_2{\mathbf{e}}^{2{\xi }_{20}}+{\delta }_1{\delta }_2{\mathbf{e}}^{2{\xi }_{10}+2{\xi }_{20}}{)}^2,\end{array}$$

(10)

(b) ${\xi }_2\sim 0,{\xi }_1\sim -\infty ,f\sim 1+b_2{\mathbf{e}}^{{\xi }_2}+{\delta }_2{\mathbf{e}}^{2{\xi }_2},$

$$\begin{array}{c}\hfill u\to u_2^{-}={\displaystyle \frac{32(1+4{\delta }_2{\mathbf{e}}^{{\xi }_{20}}+{\delta }_2{\mathbf{e}}^{2{\xi }_{20}}){\mathbf{e}}^{{\xi }_{20}}}{25{(1+{\mathbf{e}}^{{\xi }_{20}}+{\delta }_2{\mathbf{e}}^{2{\xi }_{20}})}^2}}.\end{array}$$

(11)

(2) After the action $(t\to +\infty )$,

(a) ${\xi }_1\sim 0,{\xi }_2\sim -\infty ,f\sim 1+b_1{\mathbf{e}}^{{\xi }_1}+{\delta }_1{\mathbf{e}}^{2{\xi }_1},$

$$\begin{array}{c}\hfill u\to u_1^{+}={\displaystyle \frac{2(1+4{\delta }_1{\mathbf{e}}^{{\xi }_{10}}+{\delta }_1{\mathbf{e}}^{2{\xi }_{10}}){\mathbf{e}}^{{\xi }_{10}}}{{(1+{\mathbf{e}}^{{\xi }_{10}}+{\delta }_1{\mathbf{e}}^{2{\xi }_{10}})}^2}},\end{array}$$

(12)

(b) ${\xi }_2\sim 0,{\xi }_1\sim +\infty ,f\sim b_1{\mathbf{e}}^{{\xi }_1}+{\delta }_1{\mathbf{e}}^{2{\xi }_1}+B_{12}{\mathbf{e}}^{{\xi }_1+{\xi }_2}+{\delta }_1{b}_2{a}_{12}^2{\mathbf{e}}^{2{\xi }_1+{\xi }_2}+{\delta }_2{b}_1{a}_{12}^2{\mathbf{e}}^{{\xi }_1+2{\xi }_2}+Y_{12}{\mathbf{e}}^{2{\xi }_1+2{\xi }_2},$

$$u\to u_2^{+}=2(688747536{\mathbf{e}}^{2{\xi }_{10}+{\xi }_{20}}+1076168025{\delta }_1{\mathbf{e}}^{3{\xi }_{10}}+43578162{\delta }_1{\mathbf{e}}^{3{\xi }_{10}+{\xi }_{20}}+164025{\delta }_1{\mathbf{e}}^{3{\xi }_{10}+2{\xi }_{20}}$$

$$+419904{\delta }_2{\mathbf{e}}^{2{\xi }_{10}+2{\xi }_{20}}+104976{\delta }_2{\mathbf{e}}^{2{\xi }_{10}+3{\xi }_{20}}+104976{\delta }_1^2{\mathbf{e}}^{4{\xi }_{10}+{\xi }_{20}}+1167858{\delta }_1{\delta }_2{\mathbf{e}}^{3{\xi }_{10}+2{\xi }_{20}}$$

$$+531442{\delta }_1{\delta }_2{\mathbf{e}}^{3{\xi }_{10}+3{\xi }_{20}}+419904{\delta }_1^2{\delta }_2{\mathbf{e}}^{4{\xi }_{10}+2{\xi }_{20}}+16{\delta }_1^2{\delta }_2{\mathbf{e}}^{4{\xi }_{10}+3{\xi }_{20}}+25{\delta }_1{\delta }_2^2{\mathbf{e}}^{3{\xi }_{10}+4{\xi }_{20}})/25(6561{\mathbf{e}}^{{\xi }_{10}}$$

$$\begin{array}{c}\hfill +6561{\mathbf{e}}^{{\xi }_{10}+{\xi }_{20}}+{\delta }_1{\mathbf{e}}^{2{\xi }_{10}+{\xi }_{20}}+{\delta }_2{\mathbf{e}}^{{\xi }_{10}+2{\xi }_{20}}+6561{\delta }_1{\mathbf{e}}^{2{\xi }_{10}}+{\delta }_1{\delta }_2{\mathbf{e}}^{2{\xi }_{10}+2{\xi }_{20}}{)}^2.\end{array}$$

(13)

Here, ${\xi }_{10}=x+y+{\displaystyle \frac{1}{4}}t,{\xi }_{20}={\displaystyle \frac{4}{5}}x+{\displaystyle \frac{64}{125}}y+{\displaystyle \frac{256}{3125}}t$.

In order to describe the soliton solution u of Equation (1) with Equation (8), we just focus on the parameters ${\delta }_1$ and ${\delta }_2$, while the others are fixed; that is, the independent parameters $B_{12}=b_1=b_2=k_1=1$ and $k_2={\displaystyle \frac{4}{5}}$ are taken as above. For this time, two separated solitary waves of the solution u will interplay with each other as the time increases. However, they maintain the original styles after their interactions. That is, two solitons have not changed their shapes, velocities, and amplitudes, although their phases and positions shift with the varying time. Figure 3, Figure 4 and Figure 5 show the elastic interactions of two peak solitons, two M-type solitons, and two molecular solitons with ${\delta }_1={\delta }_2={\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{150}}$, and ${\displaystyle \frac{1}{15,000}}$, respectively $(t=10)$. These plots show that the soliton structures change with the varying parameters ${\delta }_1$ and ${\delta }_2$.

We know that the breather is a periodic structure in a certain direction, but also a partially localized breathing wave. In order to derive the breather solution for the solution u of Equation (1) with Equation (8), we establish the following conjugate relation for the related parameters ($\mathrm{I}$ is a imaginary unit) [29,30,31,32]:

$$\begin{array}{c}\hfill b_1=\overline{b_2}=b_{11}+b_{22}\mathrm{I},k_1=\overline{k_2}=k+\kappa \mathrm{I},{\delta }_1=\overline{{\delta }_2}=\sigma +\rho \mathrm{I},\end{array}$$

(14)

while the others satisfy $a_{12}={\displaystyle \frac{k_1-k_2}{k_1+k_2}},Y_{12}={\delta }_1{\delta }_2{a}_{12}^4$. Therefore,

$$f\equiv f_2=1+2b_{1}cos(\Xi ){\mathbf{e}}^{\Omega }+[B_{12}-2\rho sin(2\Xi )+2\sigma cos(2\Xi )]{\mathbf{e}}^{2\Omega }$$

$$\begin{array}{c}\hfill +{\displaystyle \frac{2b_1{\kappa }^2[\rho sin(\Xi )-\sigma cos(\Xi )]{\mathbf{e}}^{3\Omega }}{k^2}}+{\displaystyle \frac{{\kappa }^4({\rho }^2+{\sigma }^2){\mathbf{e}}^{4\Omega }}{k^4}},{\xi }_1=\Xi \mathrm{I}+\Omega ,{\xi }_2=-\Xi \mathrm{I}+\Omega ,\end{array}$$

(15)

with

$$\begin{array}{c}\hfill \Xi =\kappa x+\kappa (3k^2-{\kappa }^2)y+{\displaystyle \frac{1}{4}}\kappa ({\kappa }^4-10k^2{\kappa }^2+5k^4)t,\Omega =kx+k(k^2-3{\kappa }^2)y+{\displaystyle \frac{1}{4}}k(k^4-10k^2{\kappa }^2+5{\kappa }^4)t,\end{array}$$

(16)

from Equation (8).

This transformation function can induce the breather structure for the solution u through Equation (2). For example, two kinds of breathers for the solution u of Equation (1) with Equation (15) are presented, when $B_{12}=k=1,\kappa ={\displaystyle \frac{4}{5}},\sigma =\rho ={\displaystyle \frac{1}{5}},t=0$, but $b_1=b_2=1$ or $b_1=b_2={\displaystyle \frac{1}{10}}$ (Figure 6). As the value of $b_1$ and $b_2$ varies, the breathers show different structures. Furthermore, when the module of $|{\delta }_1|$ and $|{\delta }_1| \ll {\displaystyle \frac{1}{16}},$ the breather molecules for the solution u of Equation (1) with Equation (15) can be constructed. Figure 7 owns two kinds of breather molecules, when $B_{12}=k=1,\kappa ={\displaystyle \frac{4}{5}},t=0$, but also when $b_1=b_2={\displaystyle \frac{1}{2}},\sigma =\rho ={\displaystyle \frac{1}{500}}$, or $b_1=b_2={\displaystyle \frac{1}{10}},\sigma =\rho ={\displaystyle \frac{1}{1500}}$. To a certain extent, this phenomenon shows that as the values of related parameters decrease, the distance between two lines of periodic solitons becomes larger and the breather molecule changes increasingly noticeably.

2.3. The 3-Soliton Solution

In order to derive the interaction solution of three solitons for Equation (1), we devise the auxiliary function f as follows:

$$f\equiv f_3=1+b_1{\mathbf{e}}^{{\xi }_1}+b_2{\mathbf{e}}^{{\xi }_2}+b_3{\mathbf{e}}^{{\xi }_3}+{\delta }_1{\mathbf{e}}^{2{\xi }_1}+{\delta }_2{\mathbf{e}}^{2{\xi }_2}+{\delta }_3{\mathbf{e}}^{2{\xi }_3}+B_{12}{\mathbf{e}}^{{\xi }_1+{\xi }_2}+B_{13}{\mathbf{e}}^{{\xi }_1+{\xi }_3}+B_{23}{\mathbf{e}}^{{\xi }_2+{\xi }_3}$$

$$+Y_{12}{\mathbf{e}}^{2{\xi }_1+2{\xi }_2}+Y_{13}{\mathbf{e}}^{2{\xi }_1+2{\xi }_3}+Y_{23}{\mathbf{e}}^{2{\xi }_2+2{\xi }_3}+Y_{123}{\mathbf{e}}^{2{\xi }_1+2{\xi }_2+2{\xi }_3}+{\delta }_1{b}_2{a}_{12}^2{\mathbf{e}}^{2{\xi }_1+{\xi }_2}+{\delta }_1{b}_3{a}_{13}^2{\mathbf{e}}^{2{\xi }_1+{\xi }_3}$$

$$+{\delta }_2{b}_1{a}_{12}^2{\mathbf{e}}^{{\xi }_1+2{\xi }_2}+{\delta }_2{b}_3{a}_{23}^2{\mathbf{e}}^{2{\xi }_2+{\xi }_3}+{\delta }_3{b}_1{a}_{13}^2{\mathbf{e}}^{{\xi }_1+2{\xi }_3}+{\delta }_3{b}_2{a}_{23}^2{\mathbf{e}}^{{\xi }_2+2{\xi }_3}+a_{12}{a}_{13}{a}_{23}{B}_{123}{\mathbf{e}}^{{\xi }_1+{\xi }_2+{\xi }_3}$$

$$+{\delta }_1{a}_{12}^2{a}_{13}^2{B}_{23}{\mathbf{e}}^{2{\xi }_1+{\xi }_2+{\xi }_3}+{\delta }_2{a}_{12}^2{a}_{23}^2{B}_{13}{\mathbf{e}}^{{\xi }_1+2{\xi }_2+{\xi }_3}+{\delta }_3{a}_{13}^2{a}_{23}^2{B}_{12}{\mathbf{e}}^{{\xi }_1+{\xi }_2+2{\xi }_3}$$

$$\begin{array}{c}\hfill +b_1{a}_{12}^2{a}_{13}^2{Y}_{23}{\mathbf{e}}^{{\xi }_1+2{\xi }_2+2{\xi }_3}+b_2{a}_{12}^2{a}_{23}^2{Y}_{13}{\mathbf{e}}^{2{\xi }_1+{\xi }_2+2{\xi }_3}+b_3{a}_{13}^2{a}_{23}^2{Y}_{12}{\mathbf{e}}^{2{\xi }_1+2{\xi }_2+{\xi }_3},\end{array}$$

(17)

where ${\xi }_i=k_{i}x+k_i^{3}y+{\displaystyle \frac{1}{4}}{k}_i^{5}t(i=1,2,3).$

Therefore, the constraint constants satisfy

$$\begin{array}{c}\hfill a_{ij}={\displaystyle \frac{k_i-k_j}{k_i+k_j}},Y_{ij}={\delta }_i{\delta }_j{a}_{ij}^4(1\le i<j\le 3),\end{array}$$

(18)

$$\begin{array}{c}\hfill Y_{123}={\displaystyle \frac{Y_{12}{Y}_{13}{Y}_{23}}{{\delta }_1{\delta }_2{\delta }_3}},B_{123}={\displaystyle \frac{b_3{B}_{12}}{a_{12}}}-{\displaystyle \frac{b_2{B}_{13}}{a_{13}}}+{\displaystyle \frac{b_1{B}_{23}}{a_{23}}},\end{array}$$

(19)

from the bilinear Equation (5).

To illustrate the effect of Equation (17), one can choose the independent parameters as

$$\begin{array}{c}\hfill B_{12}=B_{13}=B_{23}=b_1=b_2=b_3=k_1=1,k_2={\displaystyle \frac{4}{5}},k_3={\displaystyle \frac{3}{2}},\end{array}$$

(20)

then the expected solitons can be derived from Equation (2) through Equation (17) with Equations (18) and (19) (${\delta }_i(i=1,2,3)$ will be suitable taken). Figure 8 and Figure 9 are two typical structures for this situation $(t=5)$. Figure 8 shows the interaction of three peak solitons, while Figure 9 presents the interaction of three molecular solitons, when ${\delta }_i(i=1,2,3)={\displaystyle \frac{1}{4}}$ and ${\displaystyle \frac{1}{14,000}}$, respectively.

Finally, after the established conjugate relations for the related parameters are

$$\begin{array}{c}\hfill b_1=\overline{b_2}=b_{11}+b_{22}\mathrm{I},k_1=\overline{k_2}=k+\kappa \mathrm{I},{\delta }_1=\overline{{\delta }_2}=\sigma +\rho \mathrm{I},a_{12}=A_{12}\mathrm{I},\end{array}$$

(21)

$$\begin{array}{c}\hfill a_{13}=\overline{a_{23}}=A_{13}+A_{23}\mathrm{I},Y_{12}=y_{12},Y_{13}=\overline{Y_{23}}=y_{13}+y_{23}\mathrm{I},B_{13}=\overline{B_{23}}=b_{13}+b_{23}\mathrm{I},\end{array}$$

(22)

that is

$$\begin{array}{c}\hfill a_{12}={\displaystyle \frac{\kappa \mathrm{I}}{k}},a_{13}={\displaystyle \frac{k-k_3+\kappa \mathrm{I}}{k+k_3+\kappa \mathrm{I}}},a_{23}={\displaystyle \frac{k-k_3-\kappa \mathrm{I}}{k+k_3-\kappa \mathrm{I}}},\end{array}$$

(23)

$$\begin{array}{c}\hfill A_{12}={\displaystyle \frac{\kappa }{k}},A_{13}={\displaystyle \frac{k^2-k_3^2+{\kappa }^2}{k^2+2k{k}_3+k_3^2+{\kappa }^2}},A_{23}={\displaystyle \frac{2k_3\kappa }{k^2+2k{k}_3+k_3^2+{\kappa }^2}},\end{array}$$

(24)

$$\begin{array}{c}\hfill Y_{12}={\displaystyle \frac{{\sigma }_1{\sigma }_2{\kappa }^4}{k^4}},Y_{13}={\displaystyle \frac{{\sigma }_1{\sigma }_3{(k-k_3+\kappa \mathrm{I})}^4}{{(k+k_3+\kappa \mathrm{I})}^4}},Y_{23}={\displaystyle \frac{{\sigma }_2{\sigma }_3{(k-k_3-\kappa \mathrm{I})}^4}{{(k+k_3-\kappa \mathrm{I})}^4}},\end{array}$$

(25)

and the others satisfy Equations (16) and (19) (${\xi }_3=k_{3}x+k_3^{3}y+{\displaystyle \frac{1}{4}}{k}_3^{5}t$), we can obtain the following transformation function from Equation (17):

$$f\equiv f_3=1+b_3{\mathbf{e}}^{{\xi }_3}+{\delta }_3{\mathbf{e}}^{2{\xi }_3}+2[b_{11}cos(\Xi )-b_{22}sin(\Xi )]{\mathbf{e}}^{\Omega }+[2(\sigma cos(2\Xi )-\rho sin(2\Xi ))+B_{12}]{\mathbf{e}}^{2\Omega }$$

$$+2a_{12}^2[b_{11}(\sigma cos(\Xi )-\rho sin(\Xi ))+b_{22}(\sigma sin(\Xi )+\rho cos(\Xi ))]{\mathbf{e}}^{3\Omega }+y_{12}{\mathbf{e}}^{4\Omega }+2[b_{13}cos(\Xi )-b_{23}sin(\Xi )]{\mathbf{e}}^{\Omega +{\xi }_3}$$

$$+2{\delta }_3[(b_{22}{A}_{23}^2-2b_{11}{A}_{13}{A}_{23}-b_{22}{A}_{13}^2)sin(\Xi )+(b_{11}{A}_{13}^2-2b_{22}{A}_{13}{A}_{23}-b_{11}{A}_{23}^2)cos(\Xi )]{\mathbf{e}}^{\Omega +2{\xi }_3}$$

$$+[a_{12}(A_{13}^2+A_{23}^2)B_{123}+2b_3((\rho A_{23}^2-2\sigma A_{13}{A}_{23}-\rho A_{13}^2)sin(2\Xi )+(\sigma A_{13}^2-2\rho A_{13}{A}_{23}-\sigma A_{23}^2)cos(2\Xi ))]{\mathbf{e}}^{2\Omega +{\xi }_3}$$

$$+[{\delta }_3{B}_{12}{(A_{13}^2+A_{23}^2)}^2+2(y_{13}cos(2\Xi )-y_{23}sin(2\Xi ))]{\mathbf{e}}^{2\Omega +2{\xi }_3}+2a_{12}^2\left[\right((\sigma b_{23}-\rho b_{13})A_{13}^2-2(\sigma b_{13}+\rho b_{23})A_{13}{A}_{23}$$

$$+(\rho b_{13}-\sigma b_{23})A_{23}^2)sin(\Xi )+((\sigma b_{13}+\rho b_{23})A_{13}^2+2(\sigma b_{23}-\rho b_{13})A_{13}{A}_{23}-(\sigma b_{13}+\rho b_{23})A_{23}^2)cos(\Xi )]{\mathbf{e}}^{3\Omega +{\xi }_3}$$

$$+2a_{12}^2[((b_{22}{y}_{13}-b_{11}{y}_{23})A_{13}^2+2(b_{11}{y}_{13}+b_{22}{y}_{23})A_{13}{A}_{23}+(b_{11}{y}_{23}-b_{22}{y}_{13})A_{23}^2)sin(\Xi )+((b_{11}{y}_{13}+b_{22}{y}_{23})A_{13}^2$$

$$+2(b_{11}{y}_{23}-b_{22}{y}_{13})A_{13}{A}_{23}-(b_{11}{y}_{13}+b_{22}{y}_{23})A_{23}^2)cos(\Xi )]{\mathbf{e}}^{3\Omega +2{\xi }_3}+b_3{Y}_{12}{(A_{13}^2+A_{23}^2)}^2{\mathbf{e}}^{4\Omega +{\xi }_3}$$

$$\begin{array}{c}\hfill +Y_{123}{\mathbf{e}}^{4\Omega +2{\xi }_3}.\end{array}$$

(26)

This transformation will induce the interaction between different types of solitons. For instance, when we take the independent parameters $b_{11}=k=b_{13}=b_3=k_3=1,b_{22}=b_{23}=0,\kappa ={\displaystyle \frac{4}{5}},B_{12}={\delta }_3=2$, and $\sigma =\rho ={\displaystyle \frac{1}{5}}$, the interaction between one peak soliton and a breather for the solution u of Equation (1) with Equation (26) is presented (Figure 10a). While taking $\sigma ={\displaystyle \frac{1}{500}},\rho ={\displaystyle \frac{1}{1,5000}}$, the interaction between one peak soliton and the molecular breather for the solution u is constructed (Figure 10c).

3. Summary

In a word, the multi-soliton solutions of the (2+1)-dimensional integrable CDGKS equation are derived, which is a higher-order generalization of the Kadomtsev–Petviashvili equation. Through the Bäcklund transformation (2), the bilinear form (3) for this Equation (1) is presented. In fact, the approach through the polynomial functions, which contain the exponential functions, is applied. As a result, the lower order solutions and the related dynamic structures can be obtained. These solutions are all explicit, and the structures of solitons involve the peaks, plateaus, breathers, and their molecular structures with the suitable parameters. This methodology is rather meaningful to research an integrable nonlinear system with abundant local or nonlocal structures.

Generally speaking, a peak soliton has a sharp cusp and can model certain shallow-water wave phenomena where wave crests sharpen [33]. A plateau soliton is continuous and smooth, which can maintain a uniform amplitude over part of the wave. The M-type soliton usually indicates a multi-humped structure that can have multiple peaks that remain bound together. The breather is a periodic structure in a certain direction, but is also a partially localized breathing wave. These structures are depicted through Figure 1a,b, Figure 2a and Figure 6. The elastic interaction of two separated solitary waves will interplay with each other but maintain the original styles after their interaction. These phenomena are constructed through Figure 3, Figure 4, Figure 8 and Figure 10, while the soliton molecule is a bound state of two or more elementary solitons that travel together, with these solitons maintaining their invariant shape and distance as time increases due to the same moving velocity. These structures are shown through Figure 2b, Figure 5, Figure 7 and Figure 9.

We know that to derive the soliton molecules is one of the most important topics for an integrable system. The results derived here to form the soliton molecules are different from the velocity resonant mechanism introduced by Lou a few years ago for some integrable systems [34,35]. Under his technology, two solitons can also form such soliton molecules as the kinks, lumps, and breathers. Undoubtedly, the following ideas should guide our future work: Are there any other powerful methods for seeking the multi-soliton solutions of the (2+1)-dimensional integrable CDGKS equation? Doe other soliton molecules, such as dromions, lumps, or even compactons, exist for these solutions other than the soliton structures presented here? Could we apply this methodology to (1+1)-dimensional or (3+1)-dimensional integrable systems while retaining similar effects?