Breather wave and double-periodic soliton solutions for a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation

Yun-Xia Zhang; Li-Na Xiao

doi:10.1515/phys-2022-0058

Open Access Published by De Gruyter Open Access July 12, 2022

Breather wave and double-periodic soliton solutions for a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation

Yun-Xia Zhang and Li-Na Xiao

From the journal Open Physics

https://doi.org/10.1515/phys-2022-0058

Abstract

In this work, a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation realized to represent the propagation of unidirectional shallow water waves is investigated. We first study the breather wave solutions based on the three-wave method and the bilinear form. Second, the double-periodic soliton solutions are obtained via an undetermined coefficient method, which have not been seen in other literature. We present some illustrative figures to discuss the dynamic properties of the derived waves.

Keywords: double-periodic soliton; breather wave; three-wave method; Hirota–Satsuma–Ito equation

1 Introduction

The Hirota–Satsuma–Ito (HSI) equation emerges in the Jimbo–Miwa classification [1,2,3], which is useful in investigating the propagation of unidirectional shallow water waves [4]. Zhou and Manukure [5] presented the complexiton solutions of the HSI equation by the Hirota bilinear method and the linear superposition principle, and obtained the lump and interaction solutions to the HSI equation via the Hirota direct method [6]. Liu et al. [7] studied the N-soliton and localized wave interaction solutions of the HSI equation. Liu et al. [8] obtained the multi-wave, breather wave, and interaction solutions of the HSI equation based on the three-wave method and the homoclinic breather approach. Saima et al. [9] investigated the multiple rational rogue waves via symbolic computation approach. The aim of this work is to find the breather wave and the double-periodic soliton solutions via the undetermined coefficient method and the three-wave method.

In this article, under investigation is a (2+1)-dimensional generalized HSI equation [5]

(1) u x 3 ∫ u t d x + τ + 3 u u t + u x x t + ∫ u y t d x = 0 ,

where u = u ( x , y , t ) , τ is the arbitrary constant. Eq. (1) is the extension of the HS shallow water wave equation, which is proposed by Hirota and Satsuma via a Bäcklund transformation of the Boussinesq equation.

Making the following transformation

(2) u = 2 ( ln ξ ) x x , ξ = ξ ( x , y , t ) ,

Eq. (1) becomes

(3) ( D t D x 3 + D t D y + γ D x 2 ) ξ ⋅ ξ = − γ ξ x 2 + ξ ( γ ξ x x + ξ y t + ξ x x x t ) − 3 ξ x x t ξ x − ξ t ξ y + 3 ξ x t ξ x x − ξ t ξ x x x = 0 .

The solutions of Eq. (1) can be obtained by substituting the solutions of Eq. (3) into Eq. (2). Hence, the following work is mainly based on Eq. (3).

This article is organized as follows: Section 2 obtains the breather wave solutions by using the three-wave method; Section 3 investigates the double-periodic soliton solutions via an undetermined coefficient method; Section 4 gives a summary.

2 Breather wave solutions

Based on the three-wave method [10,11,12], we have

(4) ξ = e − t ω 3 − x ω 1 − y ω 2 + ϑ 1 e t ω 3 + x ω 1 + y ω 2 + ϑ 2 sin ( t ω 6 + x ω 4 + y ω 5 ) + ϑ 3 cos ( t ω 9 + x ω 7 + y ω 8 ) ,

where ω i ( i = 1 , 2 , … , 9 ) and ϑ i ( i = 1 , 2 , 3 ) are undetermined constants. Substituting Eq. (4) into Eq. (3), we obtain

Case I:

(5) ϑ 2 = 0 , ω 8 = ω 1 2 ( τ − 3 ω 7 ω 9 ) + ω 7 2 ( ω 7 ω 9 − τ ) + ω 3 ω 1 3 − 3 ω 3 ω 7 2 ω 1 + ω 2 ω 3 ω 9 , ω 5 = − ω 1 ω 4 ( 2 τ + 3 ω 1 ω 3 ) + ω 3 ω 4 3 − ( ω 1 3 − 3 ω 4 2 ω 1 + ω 2 ) ω 6 ω 3 , ω 2 = τ ( − ω 3 ω 1 2 − 2 ω 7 ω 9 ω 1 + ω 3 ω 7 2 ) ω 3 2 + ω 9 2 − ω 1 3 + 3 ω 7 2 ω 1 , τ = − 3 ( ω 1 2 + ω 7 2 ) ( ω 3 2 + ω 9 2 ) ( ω 7 ω 9 ϑ 3 2 + 4 ω 1 ω 3 ϑ 1 ) ( ω 3 ω 7 − ω 1 ω 9 ) 2 ( 4 ϑ 1 − ϑ 3 2 ) .

Case II:

(6) ω 6 = − ω 1 ω 3 ω 4 , ω 8 = ω 7 3 , ω 5 = ω 4 3 , τ = − 3 ω 1 ω 3 , ω 2 = − ω 1 3 , ω 9 = − ω 1 ω 3 ω 7 .

Case III:

(7) ϑ 1 = τ ω 4 2 ϑ 2 2 + τ ω 7 2 ϑ 3 2 − 4 ω 6 ω 4 3 ϑ 2 2 + ω 5 ω 6 ϑ 2 2 − 4 ω 7 3 ω 9 ϑ 3 2 + ω 8 ω 9 ϑ 3 2 4 ( τ ω 1 2 + 4 ω 3 ω 1 3 + ω 2 ω 3 ) , ω 8 = τ ω 1 2 − τ ω 7 2 + ω 3 ω 1 3 − 3 ω 7 ω 9 ω 1 2 − 3 ω 3 ω 7 2 ω 1 + ω 2 ω 3 + ω 7 3 ω 9 ω 9 , ω 5 = − ω 1 ω 4 ( 2 τ + 3 ω 1 ω 3 ) + ω 3 ω 4 3 − ( ω 1 3 − 3 ω 4 2 ω 1 + ω 2 ) ω 6 ω 3 , τ = − [ 3 ω 1 ( ω 3 2 + ω 6 2 ) ( ω 4 2 − ω 7 2 ) ( ω 3 2 + ω 9 2 ) ] / [ ( − ω 3 ω 1 2 − 2 ω 4 ω 6 ω 1 + ω 3 ω 4 2 ) ω 9 2 + 2 ω 1 ( ω 3 2 + ω 6 2 ) ω 7 ω 9 + ω 3 [ ( ω 3 ω 4 − ω 1 ω 6 ) 2 − ( ω 3 2 + ω 6 2 ) ω 7 2 ] ] , ω 2 = − τ ω 3 ω 1 2 − 2 τ ω 7 ω 9 ω 1 + τ ω 3 ω 7 2 − ω 3 2 ω 1 3 − ω 9 2 ω 1 3 + 3 ω 3 2 ω 7 2 ω 1 + 3 ω 7 2 ω 9 2 ω 1 ω 3 2 + ω 9 2 , ω 9 = ± ( ω 7 2 − ω 4 2 ) ω 3 2 + ω 6 2 ( ω 1 2 + ω 7 2 ) ω 1 2 + ω 4 2 .

Case V:

(8) ϑ 1 = ω 1 2 ϑ 2 2 − ω 1 2 ϑ 3 2 − 2 ω 7 2 ϑ 3 2 4 ω 1 2 , ω 8 = ω 1 2 ( τ − 3 ω 7 ω 9 ) + ω 7 2 ( ω 7 ω 9 − τ ) ω 9 , ω 5 = ω 1 2 ( τ − 3 ω 4 ω 6 ) + ω 4 2 ( ω 4 ω 6 − τ ) ω 6 , ω 9 = 2 τ ω 6 ω 7 2 τ ω 4 − 3 ω 6 ω 4 2 + 3 ω 6 ω 7 2 , ω 6 = ± 2 τ ω 1 3 ω 7 4 + ω 1 2 ω 7 2 , ω 2 = ω 1 ω 4 3 ω 4 − 2 τ ω 6 − ω 1 2 , ω 3 = ω 4 = 0 .

Case VI:

(9) ϑ 1 = ϑ 2 2 ( τ ω 4 2 − 4 ω 6 ω 4 3 + ω 5 ω 6 ) + ϑ 3 2 ( τ ω 7 2 − 4 ω 9 ω 7 3 + ω 8 ω 9 ) 4 τ ω 1 2 + 4 ( 4 ω 1 3 + ω 2 ) ω 3 , ω 8 = ω 1 2 ( τ − 3 ω 7 ω 9 ) + ω 7 2 ( ω 7 ω 9 − τ ) ω 9 , ω 5 = ω 1 2 ( τ − 3 ω 4 ω 6 ) + ω 4 2 ( ω 4 ω 6 − τ ) ω 6 , ω 9 = 2 τ ω 6 ω 7 2 τ ω 4 − 3 ω 6 ω 4 2 + 3 ω 6 ω 7 2 , ω 2 = ω 1 ω 4 3 ω 4 − 2 τ ω 6 − ω 1 2 , ω 3 = 0 , ω 4 = 2 τ ω 6 ± ω 6 2 ω 7 2 ( ω 1 2 + ω 7 2 ) ( 4 τ 2 + 9 ω 6 2 ( ω 1 2 + ω 7 2 ) ) ω 1 2 + ω 7 2 3 ω 6 2 .

By substituting Case I–Case VI into Eqs (2) and (4), respectively, the corresponding breather wave solutions of Eq. (1) can be obtained. In order to understand the dynamic properties of the breather wave solutions, we take the following solution corresponding to Case II as an example:

(10) u = 2 e − t ω 3 − x ω 1 + y ω 1 3 + ϑ 1 e t ω 3 + x ω 1 − y ω 1 3 − ϑ 2 sin t ω 1 ω 3 ω 4 − ω 4 ( x + y ω 4 2 ) + ϑ 3 cos t ω 1 ω 3 ω 7 − ω 7 ( x + y ω 7 2 ) [ ω 1 2 ( e − t ω 3 − x ω 1 + y ω 1 3 + ϑ 1 e t ω 3 + x ω 1 − y ω 1 3 ) + ω 4 2 ϑ 2 sin t ω 1 ω 3 ω 4 − ω 4 ( x + y ω 4 2 ) − ω 7 2 ϑ 3 cos t ω 1 ω 3 ω 7 − ω 7 ( x + y ω 7 2 ) − ω 1 ( ϑ 1 e t ω 3 + x ω 1 − y ω 1 3 − e − t ω 3 − x ω 1 + y ω 1 3 ) + ω 7 ϑ 3 sin t ω 1 ω 3 ω 7 − ω 7 ( x + y ω 7 2 ) + ω 4 ϑ 2 cos t ω 1 ω 3 ω 4 − ω 4 ( x + y ω 4 2 ) 2 / [ [ e − t ω 3 − x ω 1 + y ω 1 3 + ϑ 1 e t ω 3 + x ω 1 − y ω 1 3 − ϑ 2 sin t ω 1 ω 3 ω 4 − ω 4 ( x + y ω 4 2 ) + ϑ 3 cos t ω 1 ω 3 ω 7 − ω 7 ( x + y ω 7 2 ) 2 .

When ϑ 3 = 0 , the breather wave solution (10) has been studied in ref. [8]. When ϑ 3 ≠ 0 , Eq. (10) has not been seen in other literature. The corresponding dynamic properties are shown in Figures 1, 2, 3, 4 by selecting different values for parameters in Eq. (10).

$Figure 1 Solution (10) with ω 1 = ϑ 3 = − 1 {\omega }_{1}={{\vartheta }}_{3}=-1 , ϑ 1 = ω 7 = 2 {{\vartheta }}_{1}={\omega }_{7}=2 , ω 4 = 1 {\omega }_{4}=1 , ω 3 = 3 {\omega }_{3}=3 , ϑ 2 = 0 {{\vartheta }}_{2}=0 . (a) t = − 2 t=-2 , (b) t = 0 t=0 , and (c) t = 2 t=2 .$

Figure 1

Solution (10) with ω 1 = ϑ 3 = − 1 , ϑ 1 = ω 7 = 2 , ω 4 = 1 , ω 3 = 3 , ϑ 2 = 0 . (a) t = − 2 , (b) t = 0 , and (c) t = 2 .

$Figure 2 Solution (10) with ω 1 = ϑ 3 = − 1 {\omega }_{1}={{\vartheta }}_{3}=-1 , ϑ 1 = ω 7 = 2 {{\vartheta }}_{1}={\omega }_{7}=2 , ω 4 = 1 {\omega }_{4}=1 , ω 3 = 3 {\omega }_{3}=3 , ϑ 2 = 0 {{\vartheta }}_{2}=0 . (a) x = − 15 x=-15 , (b) x = 0 x=0 , and (c) x = 15 x=15 .$

Figure 2

Solution (10) with ω 1 = ϑ 3 = − 1 , ϑ 1 = ω 7 = 2 , ω 4 = 1 , ω 3 = 3 , ϑ 2 = 0 . (a) x = − 15 , (b) x = 0 , and (c) x = 15 .

$Figure 3 Solution (10) with ω 1 = ϑ 3 = − 1 {\omega }_{1}={{\vartheta }}_{3}=-1 , ϑ 1 = ω 7 = ϑ 2 = 2 {{\vartheta }}_{1}={\omega }_{7}={{\vartheta }}_{2}=2 , ω 4 = 1 {\omega }_{4}=1 , ω 3 = 3 {\omega }_{3}=3 . (a) t = − 2 t=-2 , (b) t = 0 t=0 , and (c) t = 2 t=2 .$

Figure 3

Solution (10) with ω 1 = ϑ 3 = − 1 , ϑ 1 = ω 7 = ϑ 2 = 2 , ω 4 = 1 , ω 3 = 3 . (a) t = − 2 , (b) t = 0 , and (c) t = 2 .

$Figure 4 Solution (10) with ω 1 = ϑ 3 = − 1 {\omega }_{1}={{\vartheta }}_{3}=-1 , ϑ 1 = ω 7 = ϑ 2 = 2 {{\vartheta }}_{1}={\omega }_{7}={{\vartheta }}_{2}=2 , ω 4 = 1 {\omega }_{4}=1 , ω 3 = 3 {\omega }_{3}=3 . (a) y = − 2 y=-2 , (b) y = 0 y=0 , and (c) y = 2 y=2 .$

Figure 4

Solution (10) with ω 1 = ϑ 3 = − 1 , ϑ 1 = ω 7 = ϑ 2 = 2 , ω 4 = 1 , ω 3 = 3 . (a) y = − 2 , (b) y = 0 , and (c) y = 2 .

3 Double-periodic soliton solutions

In ref. [13], a new ansätz function was proposed to construct double-periodic soliton structures. Subsequently, some important conclusions of nonlinear evolution equation were obtained by the new ansätz function [14,15,16]. According to the idea of refs [12,13,14], the solutions of Eq. (3) can be obtained as follows:

(11) ξ = e θ 1 [ γ 1 cos ( θ 2 ) + γ 2 sin ( θ 2 ) ] + k 1 e 2 θ 1 + e θ 3 [ γ 3 cos ( θ 4 ) + γ 4 sin ( θ 4 ) ] + k 2 e θ 4 ,

where θ i = α i x + β i y + δ i t , i = 1 , 2 , 3 , 4 and α i , β i , and δ i are unknown constants. Substituting Eq. (11) into Eq. (2) yields the double periodic soliton solutions of Eq. (1). Substituting Eq. (11) into Eq. (3), we obtain

Case (1)

(12) τ = γ 3 = δ 2 = γ 4 = β 4 = 0 , δ 1 = δ 4 , γ 2 = 6 α 2 α 1 γ 1 − 3 α 1 2 + 7 α 4 2 + 3 ( α 2 2 + α 3 2 ) , β 2 = α 2 ( α 2 2 − 3 α 1 2 ) , β 1 = − α 1 3 + 3 α 2 2 α 1 .

Case (2)

(13) α 4 = β 4 = δ 4 = γ 4 = 0 , δ 2 = − γ 2 τ ( α 2 γ 1 − α 1 γ 2 ) 2 3 α 1 ( α 1 2 + α 2 2 ) γ 1 ( γ 1 2 + γ 2 2 ) , δ 3 = − α 3 τ 3 α 1 ( α 3 − α 1 ) , γ 2 = − α 1 γ 1 α 2 , δ 1 = − α 1 2 τ 4 α 1 3 + β 1 − β 4 , β 2 = α 2 3 , β 3 = − α 3 ( 3 α 1 2 − 3 α 3 α 1 + α 3 2 ) , β 1 = − α 1 3 .

Case (3)

(14) k 1 = k 2 = γ 1 = γ 3 = 0 , β 4 = 4 α 4 3 δ 4 − α 4 2 τ δ 4 , β 2 = 4 α 2 3 δ 2 − α 2 2 τ δ 2 , δ 2 = α 2 δ 4 α 4 , δ 4 = 4 ( α 1 − α 3 ) 2 α 4 τ 3 ( α 4 4 + 2 ( ( α 1 − α 3 ) 2 − α 2 2 ) α 4 2 + ( α 2 2 + ( α 1 − α 3 ) 2 ) 2 ) , δ 3 = 4 ( α 1 − α 3 ) 3 τ 3 ( α 4 4 + 2 ( ( α 1 − α 3 ) 2 − α 2 2 ) α 4 2 + ( α 2 2 + ( α 1 − α 3 ) 2 ) 2 ) + δ 1 , β 3 = [ 4 α 1 [ α 3 ( 3 α 2 2 − α 3 2 + 3 α 4 2 ) + β 1 ] − 4 α 3 β 1 + α 1 4 − 4 α 3 α 1 3 − 6 ( α 2 2 − α 3 2 + α 4 2 ) α 1 2 + 9 α 2 4 + ( α 3 2 − 3 α 4 2 ) 2 − 6 α 2 2 ( α 3 2 + 3 α 4 2 ) ] / [ 4 ( α 1 − α 3 ) ] .

Since the formula is too long, see Appendix A for other solutions. By substituting Case (1)–Case (9) into equations (2) and (11), respectively, the corresponding double periodic soliton solutions of Eq. (1) can be derived. In order to understand the dynamic properties of the double periodic soliton solutions, we take the solution corresponding to Case (2) as an example. By selecting different values for parameters in the solution corresponding to Case (2), the dynamic properties are described in Figures 5 and 6.

$Figure 5 Solution (13) with β 1 = k 1 = − 1 {\beta }_{1}={k}_{1}=-1 , τ = α 4 = δ 4 = δ 1 = k 2 = 1 \tau ={\alpha }_{4}={\delta }_{4}={\delta }_{1}={k}_{2}=1 , α 2 = α 3 = 3 {\alpha }_{2}={\alpha }_{3}=3 , α 1 = 2 {\alpha }_{1}=2 , γ 1 = − 2 {\gamma }_{1}=-2 , γ 3 = 0 {\gamma }_{3}=0 . (a) t = − 100 t=-100 , (b) t = 0 t=0 , and (c) t = 100 t=100 .$

Figure 5

Solution (13) with β 1 = k 1 = − 1 , τ = α 4 = δ 4 = δ 1 = k 2 = 1 , α 2 = α 3 = 3 , α 1 = 2 , γ 1 = − 2 , γ 3 = 0 . (a) t = − 100 , (b) t = 0 , and (c) t = 100 .

$Figure 6 Solution (13) with β 1 = k 1 = − 1 {\beta }_{1}={k}_{1}=-1 , τ = α 4 = δ 4 = δ 1 = k 2 = 1 \tau ={\alpha }_{4}={\delta }_{4}={\delta }_{1}={k}_{2}=1 , α 2 = α 3 = 3 {\alpha }_{2}={\alpha }_{3}=3 , α 1 = 2 {\alpha }_{1}=2 , γ 1 = − 2 {\gamma }_{1}=-2 , γ 3 = 3 {\gamma }_{3}=3 . (a) t = − 10 t=-10 , (b) t = 0 t=0 , and (c) t = 10 t=10 .$

Figure 6

Solution (13) with β 1 = k 1 = − 1 , τ = α 4 = δ 4 = δ 1 = k 2 = 1 , α 2 = α 3 = 3 , α 1 = 2 , γ 1 = − 2 , γ 3 = 3 . (a) t = − 10 , (b) t = 0 , and (c) t = 10 .

4 Conclusion

In this article, the (2+1)-dimensional generalized HSI equation is studied, which is used for describing the propagation of unidirectional shallow water waves. By the three-wave method, the breather wave solutions for Eq. (1) are presented. By an undetermined coefficient method, we obtain abundant double-periodic soliton solutions, which have not been seen in other literature. The dynamic properties for these derived results are demonstrated in Figures 1–6. In Figures 1 and 2, we can observe a periodic breather wave with the change of t and x values. In Figures 3 and 4, the interaction of breather waves formed by two different periodic functions is described. In Figures 5 and 6, the double-periodic soliton structures can be found. Obviously, it is easy to obtain double periodic soliton solutions of nonlinear integrable equations using this method of undetermined coefficients. Generally speaking, if a nonlinear integrable equation has a Hirota bilinear form [17,18,19], the undetermined coefficient method can be used to obtain the double periodic soliton solutions of the equation via symbolic computation [20,21,22, 23,24,25].

Funding information: This research was funded by Jiangxi educational science “14th five year plan” project (21YB221).
Author contributions: All authors contributed to writing–original draft, methodology, software, formal analysis, and funding acquisition. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Appendix

Case (4)

(A1) k 1 = k 2 = γ 2 = γ 3 = 0 , β 4 = 4 α 4 3 δ 4 − α 4 2 τ δ 4 , β 2 = 4 α 2 3 δ 2 − α 2 2 τ δ 2 , δ 3 = [ − α 4 2 δ 2 2 τ + α 4 δ 4 δ 2 [ 2 α 2 [ 3 ( α 1 − α 3 ) δ 1 + τ ] + 3 [ − α 2 2 + ( α 1 − α 3 ) 2 + α 4 2 ] δ 2 ] + α 2 δ 4 2 [ 3 [ α 2 2 + ( α 1 − α 3 ) 2 − α 4 2 ] δ 2 − α 2 τ ] ] / [ 6 α 2 ( α 1 − α 3 ) α 4 δ 2 δ 4 ] , β 3 = [ α 4 2 δ 2 2 τ [ ( α 3 − α 1 ) [ ( α 1 − α 3 ) 2 − 3 ( α 2 2 + α 4 2 ) ] − β 1 ] + α 4 δ 4 δ 2 [ 3 δ 2 [ [ − α 2 2 + ( α 1 − α 3 ) 2 + α 4 2 ] β 1 + ( α 1 − α 3 ) [ α 1 4 − 4 α 3 α 1 3 + 2 ( α 2 2 + 3 α 3 2 − α 4 2 ) α 1 2 − 4 α 3 ( α 2 2 + α 3 2 − α 4 2 ) α 1 + 9 α 2 4 + α 3 4 − 3 α 4 4 + 2 α 2 2 α 3 2 − 2 ( 3 α 2 2 + α 3 2 ) α 4 2 ] ] + 2 α 2 τ [ ( α 3 − α 1 ) [ 2 ( α 1 − α 3 ) 2 + 3 ( α 2 2 + α 4 2 ) ] + β 1 ] ] + α 2 δ 4 2 [ 3 δ 2 [ [ α 2 2 + ( α 1 − α 3 ) 2 − α 4 2 ] β 1 + ( α 1 − α 3 ) [ α 1 4 − 4 α 3 α 1 3 + 2 ( − α 2 2 + 3 α 3 2 + α 4 2 ) α 1 2 − 4 α 3 ( − α 2 2 + α 3 2 + α 4 2 ) α 1 − 3 α 2 4 + α 3 4 + 9 α 4 4 + 2 α 3 2 α 4 2 − 2 α 2 2 ( α 3 2 + 3 α 4 2 ) ] ] + α 2 τ × [ ( α 3 − α 1 ) [ ( α 1 − α 3 ) 2 − 3 ( α 2 2 + α 4 2 ) ] − β 1 ] ] ] / [ − α 4 2 δ 2 2 τ + α 4 δ 4 δ 2 [ 3 [ − α 2 2 + ( α 1 − α 3 ) 2 + α 4 2 ] δ 2 + 2 α 2 τ ] + α 2 δ 4 2 [ 3 [ α 2 2 + ( α 1 − α 3 ) 2 − α 4 2 ] δ 2 − α 2 τ ] ] , δ 2 = ( α 2 2 δ 4 2 τ 2 ) / [ ± 2 3 α 2 2 ( α 1 − α 3 ) 2 α 4 δ 4 3 τ 2 ( τ − 3 α 4 δ 4 ) + α 2 δ 4 τ [ 3 [ α 2 2 + ( α 1 − α 3 ) 2 − α 4 2 ] δ 4 + α 4 τ ] ] .

Case (5)

(A2) k 1 = k 2 = γ 2 = γ 4 = 0 , β 4 = 4 α 4 3 δ 4 − α 4 2 τ δ 4 , β 2 = 4 α 2 3 δ 2 − α 2 2 τ δ 2 , δ 3 = 4 ( α 1 − α 3 ) 3 τ 3 ( α 4 4 + 2 ( ( α 1 − α 3 ) 2 − α 2 2 ) α 4 2 + ( α 2 2 + ( α 1 − α 3 ) 2 ) 2 ) + δ 1 , β 3 = [ 4 α 1 [ α 3 ( 3 α 2 2 − α 3 2 + 3 α 4 2 ) + β 1 ] − 4 α 3 β 1 + α 1 4 − 4 α 3 α 1 3 − 6 ( α 2 2 − α 3 2 + α 4 2 ) α 1 2 + 9 α 2 4 + ( α 3 2 − 3 α 4 2 ) 2 − 6 α 2 2 ( α 3 2 + 3 α 4 2 ) ] / [ 4 ( α 1 − α 3 ) ] , δ 2 = α 2 δ 4 α 4 , δ 4 = [ 4 ( α 1 − α 3 ) 2 α 4 τ ] / [ 3 ( α 1 2 − 2 α 3 α 1 + α 2 2 + α 3 2 + α 4 2 − 2 α 2 α 4 ) ( α 1 2 − 2 α 3 α 1 + α 2 2 + α 3 2 + α 4 2 + 2 α 2 α 4 ) ] .

Case (6)

(A3) k 1 = k 2 = γ 1 = γ 2 = 0 , β 4 = 4 α 4 3 δ 4 − α 4 2 τ δ 4 , β 2 = 4 α 2 3 δ 2 − α 2 2 τ δ 2 .

Case (7)

(A4) k 1 = γ 3 = γ 3 = 0 , β 4 = 4 α 4 3 δ 4 − α 4 2 τ δ 4 , β 2 = 4 α 2 3 δ 2 − α 2 2 τ δ 2 , τ = 3 α 2 ( α 2 2 + ( α 1 − 2 α 4 ) 2 ) δ 2 ( δ 2 2 + ( δ 1 − 2 δ 4 ) 2 ) ( ( α 1 − 2 α 4 ) δ 2 − α 2 ( δ 1 − 2 δ 4 ) ) 2 , β 1 = [ − 2 α 4 2 δ 2 τ ( γ 1 δ 1 + γ 2 δ 2 ) + 2 δ 4 2 [ δ 2 [ [ α 1 3 − 6 α 4 α 1 2 − 3 ( α 2 2 − 4 α 4 2 ) α 1 − 16 α 4 3 + 6 α 2 2 α 4 ] γ 1 + 3 α 2 [ α 2 2 + ( α 1 − 2 α 4 ) 2 ] γ 2 ] − α 2 2 γ 2 τ ] + δ 4 [ α 2 2 [ γ 2 [ 3 ( α 1 − 2 α 4 ) δ 2 2 + δ 1 τ ] + 3 ( α 1 − 2 α 4 ) γ 1 δ 1 δ 2 ] + ( α 1 − 2 α 4 ) α 2 δ 2 [ γ 2 [ − 3 ( α 1 − 2 α 4 ) δ 1 − 2 τ ] + 3 ( α 1 − 2 α 4 ) γ 1 δ 2 ] + ( α 1 − 4 α 4 ) δ 2 [ γ 1 ( α 1 ( − τ ) − ( α 1 2 − 2 α 4 α 1 + 4 α 4 2 ) δ 1 ) − ( α 1 2 − 2 α 4 α 1 + 4 α 4 2 ) γ 2 δ 2 ] + 3 α 2 3 δ 2 ( γ 1 δ 2 − γ 2 δ 1 ) ] ] / [ δ 2 δ 4 [ γ 2 δ 2 + γ 1 ( δ 1 − 2 δ 4 ) ] ] .

Case (8)

(A5) k 1 = k 2 = γ 3 = γ 4 = 0 , β 4 = 4 α 4 3 δ 4 − α 4 2 τ δ 4 , β 2 = 4 α 2 3 δ 2 − α 2 2 τ δ 2 .

Case (9)

(A6) k 1 = k 2 = γ 1 = γ 4 = 0 , β 4 = 4 α 4 3 δ 4 − α 4 2 τ δ 4 , β 2 = 4 α 2 3 δ 2 − α 2 2 τ δ 2 , δ 3 = [ α 4 2 δ 2 2 ( − τ ) + α 4 δ 4 δ 2 [ 2 α 2 [ 3 ( α 1 − α 3 ) δ 1 + τ ] + 3 [ − α 2 2 + ( α 1 − α 3 ) 2 + α 4 2 ] δ 2 ] + α 2 δ 4 2 [ 3 [ α 2 2 + ( α 1 − α 3 ) 2 − α 4 2 ] δ 2 − α 2 τ ] ] / [ 6 α 2 ( α 1 − α 3 ) α 4 δ 2 δ 4 ] , β 3 = [ α 2 ( α 1 − α 3 ) δ 4 2 τ [ ( α 1 − α 3 ) [ ( α 1 − α 3 ) 2 − 3 ( α 2 2 + α 4 2 ) ] + β 1 ] + α 2 ( α 1 − α 3 ) 2 α 4 δ 4 τ 2 − 2 3 α 3 α 1 α 2 2 ( α 1 − α 3 ) 2 α 4 δ 4 3 τ 2 ( τ − 3 α 4 δ 4 ) + 3 α 1 2 α 2 2 ( α 1 − α 3 ) 2 α 4 δ 4 3 τ 2 ( τ − 3 α 4 δ 4 ) − 3 α 2 2 α 2 2 ( α 1 − α 3 ) 2 α 4 δ 4 3 τ 2 ( τ − 3 α 4 δ 4 ) + 3 α 4 2 α 2 2 ( α 1 − α 3 ) 2 α 4 δ 4 3 τ 2 ( τ − 3 α 4 δ 4 ) + 3 α 3 2 α 2 2 ( α 3 − α 1 ) 2 α 4 δ 4 3 τ 2 ( τ − 3 α 4 δ 4 ) ] / [ α 2 ( α 1 − α 3 ) δ 4 2 τ ] , δ 2 = [ α 2 2 δ 4 2 τ 2 ] / [ 2 3 α 2 2 ( α 1 − α 3 ) 2 α 4 δ 4 3 τ 2 ( τ − 3 α 4 δ 4 ) + α 2 δ 4 τ [ 3 [ α 2 2 + ( α 1 − α 3 ) 2 − α 4 2 ] δ 4 + α 4 τ ] ] , δ 4 = 4 ( α 1 − α 3 ) 2 α 4 τ 3 ( α 4 4 + 2 ( ( α 1 − α 3 ) 2 − α 2 2 ) α 4 2 + ( α 2 2 + ( α 1 − α 3 ) 2 ) 2 ) .

References

[1] Kuo CK, Behzad G. Resonant multi-soliton solutions to new (3+1)-dimensional Jimbo–Miwa equations by applying the linear superposition principle. Nonlinear Dyn. 2019;96:459–64. 10.1007/s11071-019-04799-9Search in Google Scholar

[2] Wazwaz AM. Multiple-soliton solutions for extended (3+1)-dimensional Jimbo–Miwa equations. Appl Math Lett. 2017;64:21–6. 10.1016/j.aml.2016.08.005Search in Google Scholar

[3] Zhao ZL, He LC. M-lump and hybrid solutions of a generalized (2+1)-dimensional Hirota-Satsuma-Ito equation. Appl Math Lett. 2021;111:106612. 10.1016/j.aml.2020.106612Search in Google Scholar

[4] Ambrosinoab F, Thinováb L, Briestenskýc M, Giudicepietrod F, Rocae V, Sabbarese C. Analysis of geophysical and meteorological parameters influencing 222Rn activity concentration in Mladeč caves (Czech Republic) and in soils of Phlegrean Fields caldera (Italy). Appl Radiat Isotopes. 2020;160:109140. 10.1016/j.apradiso.2020.109140Search in Google Scholar PubMed

[5] Zhou Y, Manukure S. Complexiton solutions to the Hirota-Satsuma-Ito equation. Math Method Appl Sci. 2019;42(7):2344–51. 10.1002/mma.5512Search in Google Scholar

[6] Zhou Y, Manukure S, Ma WX. Lump and lump-soliton solutions to the Hirota-Satsuma-Ito equation. Commun Nonlinear Sci Numer Simulat. 2019;68:56–62. 10.1016/j.cnsns.2018.07.038Search in Google Scholar

[7] Liu YQ, Wen XY, Wang DS. The N-soliton solution and localized wave interaction solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation. Comput Math Appl. 2019;77(4):947–66. 10.1016/j.camwa.2018.10.035Search in Google Scholar

[8] Liu JG, Zhu WH, Zhou L. Multi-wave, breather wave, and interaction solutions of the Hirota-Satsuma-Ito equation. Eur Phys J Plus. 2020;135:20. 10.1140/epjp/s13360-019-00049-4Search in Google Scholar

[9] Saima A, Nauman R, Asma RB, Ahmad J, Aguilar JF. Multiple rational rogue waves for higher dimensional nonlinear evolution equations via symbolic computation approach. J Ocean Eng Sci. 2021. 10.1016/j.joes.2021.11.001. Search in Google Scholar

[10] Liu JG, Zhu WH. Various exact analytical solutions of a variable-coefficient Kadomtsev-Petviashvili equation. Nonlinear Dyn. 2020;100:2739–51. 10.1007/s11071-020-05629-zSearch in Google Scholar

[11] Liu JG, Zhu WH. Breather wave solutions for the generalized shallow water wave equation with variable coefficients in the atmosphere, rivers, lakes and oceans. Comput Math Appl. 2019;78:848–56. 10.1016/j.camwa.2019.03.008Search in Google Scholar

[12] Liu JG, Zhu WH. Multiple rogue wave, breather wave and interaction solutions of a generalized (3+1)-dimensional variable-coefficient nonlinear wave equation. Nonlinear Dyn. 2021;103:1841–50. 10.1007/s11071-020-06186-1Search in Google Scholar

[13] Long W. Multiple periodic-soliton solutions to Kadomtsev-Petviashvili equation. Appl Math Comput. 2011;218:368–75. 10.1016/j.amc.2011.05.072Search in Google Scholar

[14] Liu JG. Double-periodic soliton solutions for the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid. Comput Math Appl. 2018;75:3604–13. 10.1016/j.camwa.2018.02.020Search in Google Scholar

[15] Liu JG, Zhu WH, Lei ZQ, Ai GP. Double-periodic soliton solutions for the new (2+1)-dimensional KdV equation in fluid flows and plasma physics. Anal Math Phys. 2020;10:41. 10.1007/s13324-020-00387-ySearch in Google Scholar

[16] Liu JG, Tian Y. New double-periodic soliton solutions for the (2+1)-dimensional breaking soliton equation. Commun Theor Phys. 2018;69;585–97. 10.1088/0253-6102/69/5/585Search in Google Scholar

[17]