Abundant exact solutions of higher-order dispersion variable coefficient KdV equation

Zhen Zhao; Jing Pang

doi:10.1515/phys-2022-0190

Open Access Published by De Gruyter Open Access September 27, 2022

Abundant exact solutions of higher-order dispersion variable coefficient KdV equation

Zhen Zhao and Jing Pang

From the journal Open Physics

https://doi.org/10.1515/phys-2022-0190

Abstract

In this article, various exact solutions of the fifth-order variable coefficient KdV equation with higher-order dispersion term are studied. Because of the complexity of the exact solution of the variable coefficient t, it has a certain influence on the tension waves at the fluid interface on the gravity surface. First, the bilinear KdV equation is derived by using the Hirota bilinear method, and four mixed solutions consisting of positive quartic function, quadratic function, exponential function, and hyperbolic function are constructed. Second, the linear superposition principle is used to obtain the resonance multisoliton solution, and two cases are taken as examples to illustrate the study of resonance multi soliton solution. In addition, 3D images and contour images are drawn by mathematical symbol calculation and appropriate parameters, and the process of tension fluctuation is vividly explained by physical phenomena. The results obtained greatly expand the exact solution of the KdV equation in the existing literature and enable us to understand nonlinear dynamical systems more deeply.

Keywords: tension waves; fifth order variable coefficient KdV equation; Hirota bilinear method; linear superposition principle; interaction solution; resonant multi soliton solution

1 Introduction

Onlinear partial differential equations [1, 2,3,4,5] have been widely used in natural and social sciences, especially in the fields of plasma physics, ocean dynamics, lattice dynamics, and hydrodynamics [6]. The most famous non linear variable coefficient partial differential equation is the variable coefficient KdV equation. Until now, many scholars have done a lot of research on the complex non linear waves of nonlinear constant coefficient and variable coefficient KdV equations, and put forward various effective methods to solve their exact solutions and numerical solutions, for example, Bäcklund transform method [7,8], Hirota bilinear method [9,10,11], ( G ′ ∕ G ) expansion method [12], multisoliton solution [13], Wronskian method [14], Painlevé analysis [15], and other methods.

At the same time, it is found that the physical properties described by various types of KdV equations are different. For example, in ref. [16], the propagation characteristics of nonlinear solitary waves in the ocean were described, and the effects of dissipation term and perturbation term contained in KdV on the elastic collision of nonlinear solitary waves in two oceans were given. In ref. [17], unstable drift waves in plasma physics were described. In ref. [18], the propagation of solitons in inhomogeneous propagation media in the fields of quantum mechanics and nonlinear mechanics was described. In ref. [19], nonlinear solitary wave fluctuations in the atmosphere and ocean with large amplitude were described. Reference [20] showed that by simulating the propagation depth and width of small amplitude surface waves in large channels and straits, the transformation was slow, and the vorticity did not disappear. After more in-depth study of nonlinear partial differential equations with variable coefficients, scholars found that the coefficient of the dissipation term of the variable coefficient KdV equation changes with time and space, and the variable coefficient KdV equation can better describe the physical phenomena and physical properties behind it. Therefore, the research on the variable coefficient KdV equation has increased in recent years, and it is found that the aforementioned methods for studying the constant coefficient KdV equation can also be used to study the variable coefficient KdV equation, and some progress has been made. For example, in ref. [21], the author used pfaffians form to represent the multi soliton solution of the variable coefficient coupled KdV equation and further analyzed the dynamic characteristics of the soliton solution. In ref. [22], the author used the improved sine cosine method to construct the exact periodic solutions and soliton solutions of two groups of the fifth-order KdV equations with variable coefficients and linear damping term. In ref. [23], the author first obtained the N-soliton solution of the (2 + 1)-dimensional KdV equation with variable coefficients by using the bell polynomial method and analyzed the influence of soliton fission, fission and rear end collision on the coefficients, obtained the Bäcklund transformation of the equation, and then obtained the periodic wave solution of the equation by using the Riemann function method.

In this article, the generalized variable coefficient KdV equation with higher-order dissipative term is studied:

(1) u t + a ( t ) u u x + b ( t ) u x x x + c ( t ) u 2 u x + d ( t ) u x u 2 x + e ( t ) u u x x x + f ( t ) u x x x x x + g ( t ) u x ∂ x − 1 u y + h ( t ) u x x y + k ( t ) u u y = 0 ,

u t + a ( t ) u u x + b ( t ) u x x x + c ( t ) u 2 u x + d ( t ) u x u 2 x + e ( t ) u u x x x + f ( t ) u x x x x x + g ( t ) u x v + h ( t ) v x x x + k ( t ) u v x = 0 , u y = v x ,

where u = u ( x , y , t ) in Eq. (1) is the function space of x , y , t . a ( t ) , c ( t ) , d ( t ) , e ( t ) , g ( t ) , and k ( t ) are nonlinear term of time t , b ( t ) , and h ( t ) represent linear dispersion of time t , and f ( t ) represents higher-order dispersion on time t . Eq. (1) shows tension waves on a gravitational surface at a fluid interface, where surface tension, gravity, and fluid inertia are affected.

Eq. (1) is the interaction between solitons, and the interaction between solitons can be divided into elastic and inelastic collisions. The variable coefficients f ( t ) and l ( t ) of the union Eq. (1) can lead to fusion and fission of solitons.

This article is organized as follows. In Section 2, the bilinear form of the KdV equation with a variable coefficients is obtained by using the Hirota bilinear method. In Section 3, by constructing the combined solutions of positive quartic function, quadratic function, exponential function, and hyperbolic function, the three-dimensional space corresponding to the interaction solutions of high-order Lump solution and periodic cross kink solution is obtained, and the forms of accurate solutions are more abundant. The shape and trajectory of waves are vividly displayed through Maple. In Section 4, the resonant multi soliton solution of the KdV equation with variable coefficients is obtained by linear superposition principle, and the physical structure of tension waves is illustrated by N = 2 , N = 3 , and N = 6 as examples. The symbolic calculation is carried out by using mathematical software, and the high-order lumps solution, the interaction solution with periodic cross kink solution, and resonant multiple solitons are obtained, which supplements the research on the exact solution of the generalized variable coefficient fifth-order variable coefficient KdV equation. The fifth part is the summary.

2 Preparatory theory

When c ( t ) = δ , d ( t ) = γ , e ( t ) = β , f ( t ) = α , and a ( t ) = g ( t ) = h ( t ) = k ( t ) = 0 , Eq. (1) transforms into the generalized fifth-order KdV equation [24] with constant coefficients,
(2) u t + α u x x x x x + β u u x x x + γ u x u x x + δ u 2 u x = 0 .
By using the Ansatz and Jacobi elliptic function expansion method, the elliptic cosine wave solution of the equation is constructed.
When a ( t ) = b ( t ) = g ( t ) = h ( t ) = k ( t ) = 0 , Eq. (1) transforms into variable coefficient Sawada-Kotera equation [25],
(3) u t + f ( t ) u 2 u x + g ( t ) u x u x 2 + h ( t ) u u x 3 + k ( t ) u x 5 = 0 .
The auto-Bäcklund transform, soliton solution, and random soliton solution are obtained by using Hermite transform in Kondrativ distribution space.
When k ( t ) = 4 h ( t ) = 2 g ( t ) , a ( t ) = 6 b ( t ) , and c ( t ) = d ( t ) = e ( t ) = f ( t ) = 0 , Eq. (1) is transformed into a generalized (2 + 1)-dimensional equation [26],
(4) u t − h 1 ( 4 u u y + 2 u x ∂ x − 1 u y + u x x y ) − h 2 ( 6 u u x + u x x x ) = 0 .

The bilinear form, bilinear Bäcklund transformation, Lax pair, and Darboux transformation are constructed by using binary Bell polynomials. The equation is simplified into integrable equation, and the infinite conservation law of the equation is obtained by using binary Bell polynomials.

In this section, the Hirota bilinear method is used to obtain the bilinear form of Eq. (1), and then the form of solution is constructed to obtain the interaction solution of the equation.

First, the transformation of related variables is used:

(5) u = 2 ( ln g ) x x .

Convert Eq. (1) into the bilinear equation as follows:

(6) B KdV ( g ) = ( D x D t + b ( t ) D x 4 + f ( t ) D x 6 + h ( t ) D x 3 D y ) g ⋅ g = − g t g x + g g x t + 3 b ( t ) g x x 2 − 4 b ( t ) g x g x x x − 10 f ( t ) g x x x 2 + b ( t ) g g x x x x + 15 f ( t ) g x x g x x x x − 6 f ( t ) g x g x x x x x + f ( t ) g g x x x x x x + h ( t ) g g x x x y − 3 h ( t ) g x g x x y + 3 h ( t ) g x x g x y − h ( t ) g x x x g y = 0 ,

where g = g ( x , y , t ) , and D -operator is defined by ref. [27]:

(7) D x m D y n D t k f g = ∂ ∂ x − ∂ ∂ x ′ m ∂ ∂ y − ∂ ∂ y ′ n ∂ ∂ t − ∂ ∂ t ′ k f ( x , y , t ) g ( x ′ , y ′ , t ′ ) ∣ x ′ = x , y ′ = y , t ′ = t .

2.1 Interaction solution

To obtain the interaction solution between the high-order Lump solution and the periodic cross kinking solution of the fifth-order variable coefficient KdV equation, the following positive quartic function, quadratic function, and the test function of the combination of exponential function and hyperbolic function are assumed,

(8) g = e − p 1 ξ 1 + k 1 e p 1 ξ 2 + k 2 cos ( p 2 ξ 2 ) + k 3 cosh ( ξ 3 ) + k 4 sinh ( ξ 4 ) + k 5 sin ( ξ 5 ) + a 21 ,

where

ξ 1 = a 1 x + a 2 y + a 3 ( t ) + a 4 , ξ 2 = a 5 x + a 6 y + a 7 ( t ) + a 8 , ξ 3 = a 9 x + a 10 y + a 11 ( t ) + a 12 , ξ 4 = a 13 x + a 14 y + a 15 ( t ) + a 16 , ξ 5 = a 17 x + a 18 y + a 19 ( t ) + a 20 .

where a i ( 2 ≤ i ≤ 21 ) , k j ( 1 ≤ j ≤ 5 ) are constant. A set of algebraic equations about a i ( 1 ≤ i ≤ 21 , and i ≠ 3 , 7 , 9 , 11 , 15 , 19 ) are obtained by substituting Eq. (8) into Eq. (1) and making the coefficients of x and y zero. By solving the algebraic equations, we can find the following groups of rich interaction solutions.

Case 1:

(9) a 1 = 0 , a 3 ( t ) = a 3 ( t ) , a 5 = 0 , a 9 = 0 , a 13 = 0 , a 15 ( t ) = ∫ a t a 17 2 a 14 h ( s ) d s , p 1 = 0 , f ( t ) = f ( t ) , a 19 ( t ) = ∫ a t 4 a 17 2 f ( s ) d s , h ( t ) = h ( t ) , k 1 = 0 , a 11 ( t ) = ∫ a t a 17 2 a 10 h ( s ) d s , k 2 = 0 .

a 2 , a 4 , a 6 , a 8 , a 10 , a 12 , a 14 , a 16 , a 18 , k 3 , k 4 , k 5 , and p 2 are all constants. a 17 is a constant that is not equal to zero, and the relation between the dispersion term of Eq. (1) and the higher-order dispersion term is b ( t ) = 5 a 17 3 f ( t ) − a 18 h ( t ) a 17 . The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(10) u 1 ( x , y , t ) = Φ 1 Ψ 1 ,

where

Φ 1 = − 2 a 17 2 k 5 ( sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) × ( k 4 sinh ( a 15 ( t ) + a 14 y + a 16 ) + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) + k 5 ) , Ψ 1 = 2 k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) ( k 4 sinh ( a 15 ( t ) + a 14 y + a 16 ) + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) + 2 ( a 21 + 1 ) k 4 sinh ( a 15 ( t ) + a 14 y + a 16 ) + k 3 2 cosh ( a 11 ( t ) + a 10 y + a 12 ) 2 + k 4 2 cosh ( a 15 ( t ) + a 14 y + a 16 ) 2 + ( a 21 + 1 ) 2 + 2 k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) ( a 21 + 1 + k 4 sinh ( a 15 ( t ) + a 14 y + a 16 ) ) − k 5 2 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) 2 − k 4 2 + k 5 2 .

Case 2:

(11) a 1 = 0 , a 3 ( t ) = ∫ a t a 17 2 a 2 h ( s ) d s , a 5 = 0 , a 7 ( t ) = a 7 ( t ) , a 11 ( t ) = ∫ a t a 17 2 a 10 h ( s ) d s , a 13 = 0 , a 15 ( t ) = ∫ a t a 17 2 a 14 h ( s ) d s , k 2 = 0 , a 19 ( t ) = ∫ a t 4 a 17 5 f ( s ) d s , a 9 = 0 , f ( t ) = f ( t ) , k 1 = 0 , h ( t ) = h ( t ) .

a 2 , a 4 , a 6 , a 8 , a 10 , a 12 , a 14 , a 16 , a 18 , k i ( 3 ≤ i ≤ 5 ) , p 1 , and p 2 are all constants. a 17 is a constant that is not equal to zero, and the relation between the dispersion term of Eq. (1) and the higher-order dispersion term is b ( t ) = 5 a 17 3 f ( t ) − a 18 h ( t ) a 17 . The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(12) u 2 ( x , y , t ) = Φ 2 Ψ 2 ,

where

Φ 2 = − 2 a 17 2 k 5 ( sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) ( k 4 sinh ( a 15 ( t ) + a 14 y + a 16 ) + s i n ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) e − p 1 ( a 3 ( t ) + a 2 y + a 4 ) + k 5 + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 ) ) , Ψ 2 = e − p 1 ( a 3 ( t ) + a 2 y + a 4 ) ( 2 k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + 2 k 4 sinh ( a 15 ( t ) + a 14 y + a 16 ) + 2 k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + 2 a 21 ) − k 5 2 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) 2 + sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) ( a 21 2 − k 4 2 + 2 k 4 k 5 sinh ( a 15 ( t ) + a 14 y + a 16 ) + k 5 2 + 2 a 21 k 5 + 2 k 3 k 5 cosh ( a 11 ( t ) + a 10 y + a 12 ) ) + 2 a 21 k 4 sinh ( a 15 ( t ) + a 14 y + a 16 ) + k 3 2 cosh ( a 11 ( t ) + a 10 y + a 12 ) 2 + k 4 2 cosh ( a 15 ( t ) + a 14 y + a 16 ) 2 ( 2 k 4 k 3 sinh ( a 15 ( t ) + a 14 y + a 16 ) + 2 a 21 k 3 ) + cosh ( a 11 ( t ) + a 10 y + a 12 ) + e − 2 p 1 ( a 3 ( t ) + a 2 y + a 4 ) .

Case 3:

(13) a 1 = 0 , a 3 ( t ) = a 3 ( t ) , a 5 = 0 , a 7 ( t ) = a 7 ( t ) , a 9 = 0 , k 2 = 0 , a 11 ( t ) = − ∫ a t a 13 2 a 10 h ( s ) d s , a 15 ( t ) = 4 ∫ a t a 13 5 f ( s ) d s , a 17 = 0 , f ( t ) = f ( t ) , a 19 ( t ) = − ∫ a t a 13 2 a 18 h ( s ) d s , h ( t ) = h ( t ) , p 1 = 0 , k 1 = 0 .

a 2 , a 4 , a 6 , a 8 , a 10 , a 12 , a 13 , a 14 , a 16 , a 18 , k i ( 3 ≤ i ≤ 5 ) , and p 2 are all constants. a 13 is a constant that is not equal to zero, and the relation between the dispersion term of Eq. (1), and the higher-order dispersion term is b ( t ) = − 5 a 13 3 f ( t ) + a 14 h ( t ) a 13 . The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(14) u 3 ( x , y , t ) = Φ 3 Ψ 3 ,

where

Φ 3 = 2 a 13 2 k 4 ( k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) 2 − k 4 cosh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) 2 + sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) ( k 5 sin ( a 19 ( t ) + a 18 y + a 20 ) + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) ) , Ψ 3 = ( k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) + k 5 sin ( a 19 ( t ) + a 18 y + a 20 ) + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) 2 .

Case 4:

(15) a 1 = 0 , a 3 ( t ) = − ∫ a t a 13 2 a 2 h ( s ) d s , a 5 = 0 , a 9 = 0 , a 11 ( t ) = − ∫ a t a 13 2 a 10 h ( s ) d s , h ( t ) = h ( t ) , a 15 ( t ) = 4 ∫ a t a 13 5 f ( s ) d s , a 17 = 0 , f ( t ) = f ( t ) , a 19 ( t ) = − ∫ a t a 13 2 a 18 h ( s ) d s , k 1 = 0 , k 2 = 0 .

a 2 , a 4 , a 6 , a 8 , a 10 , a 12 , a 13 , a 14 , a 16 , a 18 , k i ( 3 ≤ i ≤ 5 ) , p 1 , and p 2 are all constants. a 13 is a constant that is not equal to zero, and the relation between the dispersion term of Eq. (1) and the higher-order dispersion term is b ( t ) = − 5 a 13 3 f ( t ) + a 14 h ( t ) a 13 . The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(16) u 4 ( x , y , t ) = Φ 4 Ψ 4 ,

where

Φ 4 = 2 a 13 2 k 4 ( k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) 2 − k 4 cosh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) 2 + sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) ( k 5 sin ( a 19 ( t ) + a 18 y + a 20 ) + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 ) + e − p 1 ( a 3 ( t ) + a 2 y + a 4 ) sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) ) , Ψ 4 = ( k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) + k 5 sin ( a 19 ( t ) + a 18 y + a 20 ) + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + e − p 1 ( a 3 ( t ) + a 2 y + a 4 ) ) 2 .

Case 5:

(17) a 1 = 0 , a 3 ( t ) = a 3 ( t ) , a 5 = 0 , a 7 ( t ) = a 7 ( t ) , a 9 = 0 , a 10 = 0 , a 11 ( t ) = 0 , a 14 = a 13 a 18 a 17 , a 15 ( t ) = 0 , a 19 ( t ) = 0 , k 1 = 0 , k 2 = 0 , p 1 = 0 , f ( t ) = 0 , h ( t ) = h ( t ) .

a 2 , a 4 , a 6 , a 8 , a 12 , a 13 , a 16 , a 18 , a 20 , a 21 , k i ( 3 ≤ i ≤ 5 ) , and p 2 are all constants. a 17 is not equal to zero for the constant, the high-order dispersion term of Eq. (1) is zero, and the relationship between a linear dispersion b ( t ) = − a 18 h ( t ) a 17 . The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(18) u 5 ( x , y , t ) = Φ 5 Ψ 5 ,

where

Φ 5 = − 2 a 17 2 k 5 ( k 3 cosh ( a 11 ( t ) + a 12 ) + a 21 + 1 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) − ( a 17 2 − a 13 2 ) k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) 2 k 4 sinh a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 − a 13 2 ( k 3 cosh ( a 11 ( t ) + a 12 ) + a 21 + 1 ) − 4 a 13 a 17 k 4 k 5 cos ( a 19 ( t ) − 2 a 17 2 k 5 2 − 2 a 13 2 k 4 2 + a 17 x + a 18 y + a 20 ) cosh a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 , Ψ 5 = 2 k 5 ( k 3 cosh ( a 11 ( t ) + a 12 ) + a 21 + 1 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + 2 k 4 sinh ( a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 ( k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + k 3 cosh ( a 11 ( t ) + a 12 ) + a 21 + 1 ) + ( a 21 + 1 ) 2 + 2 ( a 21 + 1 ) k 3 cosh ( a 11 ( t ) + a 12 ) + k 3 2 cosh ( a 11 ( t ) + a 12 ) 2 − k 4 2 + k 5 2 + k 4 2 cosh a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 2 − k 5 2 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) 2 .

Case 6:

(19) a 1 = 0 , a 2 = 0 , a 3 ( t ) = 0 , a 5 = 0 , a 7 ( t ) = a 7 ( t ) , a 9 = 0 , a 10 = 0 , a 11 ( t ) = 0 , a 14 = a 13 a 18 a 17 , a 15 ( t ) = 0 , a 19 ( t ) = 0 , k 1 = 0 , k 2 = 0 , f ( t ) = 0 , h ( t ) = h ( t ) .

a 4 , a 6 , a 8 , a 12 , a 13 , a 16 , a 18 , a 20 , a 21 , k i ( 3 ≤ i ≤ 5 ) , p 1 , and p 2 are all constants. a 17 is not equal to zero for the constant, the high-order dispersion term of Eq. (1) is zero, and the relationship between a linear dispersion b ( t ) = − a 18 h ( t ) a 17 . The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(20) u 6 ( x , y , t ) = Φ 6 Ψ 6 ,

where

Φ 6 = − 2 a 17 2 k 5 ( k 3 cosh ( a 11 ( t ) + a 12 ) + e − p 1 ( a 3 ( t ) + a 4 ) + a 21 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + ( a 13 2 − a 17 2 ) k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + 2 k 4 sinh ( a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 ) ( e − p 1 ( a 3 ( t ) + a 4 ) + a 13 2 ( k 3 cosh ( a 11 ( t ) + a 12 ) + a 21 ) ) − 4 a 13 a 17 k 4 k 5 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) cosh ( a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 ) − 2 a 17 2 k 5 2 − 2 a 13 2 k 4 2 ,

Ψ 6 = 2 k 5 ( k 3 cosh ( a 11 ( t ) + a 12 ) + e − p 1 ( a 3 ( t ) + a 4 ) + a 21 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + 2 k 4 sinh a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 ( k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + k 3 cosh ( a 11 ( t ) + a 12 ) + e − p 1 ( a 3 ( t ) + a 4 ) + a 21 ) + e − p 1 ( a 3 ( t ) + a 4 ) ( 2 k 3 cosh ( a 11 ( t ) + a 12 ) + 2 a 21 ) − k 5 2 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) 2 + 2 a 21 k 3 cosh ( a 11 ( t ) + a 12 ) + k 3 2 cosh ( a 11 ( t ) + a 12 ) 2 + e − 2 p 1 ( a 3 ( t ) + a 4 ) + a 21 2 − k 4 2 + k 5 2 + k 4 2 cosh ( a 15 ( t ) + ( a 13 x + a 16 ) + a 13 a 18 y a 17 ) 2 .

Case 7:

(21) a 1 = 0 , a 3 ( t ) = a 3 ( t ) , a 5 = 0 , a 7 ( t ) = a 7 ( t ) , a 9 = 0 , a 11 ( t ) = 0 , a 15 ( t ) = 0 , a 19 ( t ) = 0 , k 1 = 0 , k 2 = 0 , p 1 = 0 , f ( t ) = 0 , h ( t ) = 0 , b ( t ) = 0 .

a 2 , a 4 , a 6 , a 8 , a 10 , a 12 , a 13 , a 14 , a 16 , a 18 , a 20 , a 21 , k i ( 3 ≤ i ≤ 5 ) , and p 2 are all constants. Both the dispersion term and the higher-order dispersion term of Eq. (1) are zero. The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(22) u 7 ( x , y , t ) = Φ 7 Ψ 7 ,

where

Φ 7 = − 4 a 13 a 17 k 4 k 5 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) cosh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) − 2 a 17 2 k 5 ( k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) − ( x + a 14 y + a 16 ) − 2 k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) ( ( a 17 2 − a 13 2 ) k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) − a 13 2 ( k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) ) − 2 a 17 2 k 5 2 − 2 a 13 2 k 4 2 ,

Ψ 7 = − k 5 2 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) 2 + k 4 2 cosh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) 2 + 2 k 5 ( k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + 2 k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) ( k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 + 1 ) + 2 ( a 21 + 1 ) k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + k 3 2 cosh ( a 11 ( t ) + a 10 y + a 12 ) 2 + ( a 21 + 1 ) 2 − k 4 2 + k 5 2 .

Case 8:

(23) a 1 = 0 , a 3 ( t ) = a 3 ( t ) , a 5 = 0 , a 7 ( t ) = a 7 ( t ) , a 9 = 0 , a 11 ( t ) = 0 , a 15 ( t ) = 0 , a 19 ( t ) = 0 , k 1 = 0 , k 2 = 0 , f ( t ) = 0 , h ( t ) = 0 , b ( t ) = 0 .

a 2 , a 4 , a 6 , a 8 , a 10 , a 12 , a 13 , a 14 , a 16 , a 17 , a 18 , a 20 , a 21 , k i ( 3 ≤ i ≤ 5 ) , p 1 , and p 2 are all constants. Both the dispersion term and the higher-order dispersion term of Eq. (1) are zero. The interaction solution of Eq. (1) can be obtained by transformation Eq. (5).

(24) u 8 ( x , y , t ) = Φ 8 Ψ 8 ,

where

Φ 8 = e − p 1 ( a 3 ( t ) + a 2 y + a 4 ) ( 2 a 13 2 k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) − 2 a 17 2 k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) ) − 4 a 13 a 17 k 4 k 5 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) cosh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) − 2 a 17 2 k 5 ( k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + 2 k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) ( ( a 13 2 − a 17 2 ) k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + a 13 2 ( k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + a 21 ) ) − 2 a 17 2 k 5 2 − 2 a 13 2 k 4 2 ,

Ψ 8 = e − p 1 ( a 3 ( t ) + a 2 y + a 4 ) ( 2 k 5 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + a 21 2 + 2 k 4 sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) + 2 k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + 2 a 21 ) − k 5 2 cos ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) 2 + k 4 2 cosh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) 2 + ( 2 k 3 k 5 cosh ( a 11 ( t ) + a 10 y + a 12 ) + 2 a 21 k 5 ) sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + sinh ( a 15 ( t ) + a 13 x + a 14 y + a 16 ) ( 2 k 5 k 4 sin ( a 19 ( t ) + a 17 x + a 18 y + a 20 ) + 2 k 3 k 4 cosh ( a 11 ( t ) + a 10 y + a 12 ) + 2 a 21 k 4 ) + 2 a 21 k 3 cosh ( a 11 ( t ) + a 10 y + a 12 ) + k 3 2 cosh ( a 11 ( t ) + a 10 y + a 12 ) 2 − k 4 2 + k 5 2 + e − 2 p 1 ( a 3 ( t ) + a 2 y + a 4 ) .

Case 9:

(25) a 1 = 0 , a 3 ( t ) = a 3 ( t ) , a 5 = 0 , a 7 ( t