# New Generalized Soliton Solutions for a (3+1)-Dimensional Equation

Authors: Yiren Chen
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we investigate the nonlinear wave solutions for a (3+1)-dimensional equation which can be reduced to the potential KdV equation. We present generalized N-soliton solutions in which some arbitrarily differentiable functions are involved by using a simplified Hirota’s method. Our work extends some previous results.


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## Body

## 1. Introduction

The research of the exact solutions for nonlinear equations plays a significant role in the investigation of nonlinear science. For nonlinear science, owing to containing more information and describing many natural phenomena, such as plasmas, gas dynamics, fluid mechanics, and elastic media, many nonlinear equations have been investigated [1–10]. Compared with the lower-dimensional nonlinear equations, the higher-dimensional nonlinear equations are more difficult to study because of the complex construction and multiple variables. Clarkson and Mansfield [3] introduced the following equation:
(1)uxxxt+βuxuxt+utuxx−uxx+uxt=0,which they called the SWWI equation, and another equation
(2)uxxxt+β2uxuxt+utuxx−uxx+uxt=0,which they called the SWWII equation.For Equation (1), Clarkson and Mansfield [3] obtained an interesting solution
(3)ux,t=3βtanhx+ft2+3βtanhx−ft2+tβ,which contains an arbitrarily differentiable function ft. For Equation (2), it seems that there is no solution which is similar to (3). Several other equations and their multiple soliton solutions were investigated by Tian and Zhang [11, 12], Yue and Chen [13], and Wazwaz [14, 15]. Furthermore, the other new and interesting solutions of some important nonlinear equations were investigated, such as the NlSm,n equation [16], the GPm,n equation [17], and new nonlocal models [18].Based on Equations (1) and (2), a (3+1)-dimensional equation
(4)uyzt+uxxxyz−6uxuxyz−6uxyuxz=0was introduced by Wazwaz [19] as a higher-dimensional shallow water wave equation. It is easy to see that Equation (4) can be reduced to the potential KdV equation for z=y=x.In [19], Wazwaz investigated multiple soliton solutions and multiple singular soliton solutions of Equation (4). For Equation (4), Wazwaz gave the solutions combined by ekix+riy+siz−ki3ti=1,2,3, where ki, ri, and si are arbitrary constants. In [20], we succeed in constructing the following generalized soliton solutions:
(1)
The generalized 1-soliton solution(5)u=−2k1φ1eθ1py,z+φ1eθ1+γt,y,z
(2)
The generalized 2-soliton solution(6)u=−2k1φ1eθ1+k2φ2eθ2+a0k1+k2φ1φ2eθ1+θ2py,z+φ1eθ1+φ2eθ2+a0φ1φ2eθ1+θ2+γt,y,z,where k1 and k2 are arbitrary constants; gt,y, ht,z, ey,z, py,z, φ1=φ1y,z, and φ2=φ2y,z are arbitrarily differentiable functions; γt,y,z=gt,y+ht,z+ey,z; and a0=k1−k22/k1−k22py,zHowever, we failed to gain the generalizedN-soliton solutions N≥3 with the same form.In this paper, using a simplified Hirota’s method [21] which is different from classical Hirota’s method [22–25], we obtain generalized N-soliton solutions and their dispersion relations. For Equation (4), these solutions are combined by py,z and φiy,zekix−ki3t, where φiy,zi=1,2,⋯,n are arbitrarily differentiable functions and kii=1,2,⋯,n are arbitrary constants. This construction of N-soliton solution is novel and has not been seen before.The structure of this paper is organized as follows: In Section2, our main results are presented. A short conclusion is given in Section 3.


## 2. The New GeneralizedN-Soliton Solutions

In this section, we state the main results for Equation (4) by means of simplified Hirota’s method [21]. Depending on the different constructions of Equation (4), we use more complex test functions.Firstly, we consider the linear part of Equation (4)
(7)uyzt+uxxxyz=0.Substitutingu=φiekix−cit (i=1,2,⋯,n and ci should be determined) into Equation (7), we obtain ci=ki3.ForN=1, based on the different constructions of Equation (4) and ci=ki3, we can let
(8)F1=py,z+φ1eθ1and assume that Equation (4) has solution of the following form:
(9)u=R∂F1/∂xF1+γt,y,z,where R is to be determined. Substituting (9) into Equation (4), we acquire R=−2 and the following generalized 1-soliton solution:
(10)u=−2k1φ1eθ1py,z+φ1eθ1+γt,y,z,which is equal to Equation (5).ForN=2, let
(11)F2=py,z+φ1eθ1+φ2eθ2+φ3eθ3+apy,zφ1φ2eθ1+θ2,where a is to be determined. Because of R=−2, we can assume that Equation (4) has solution of the following form:
(12)u=−2∂F2/∂xF2+γt,y,z.Substituting (12) into Equation (4), then we obtain a=k1−k22/k1−k22, a0=a/py,z, and the following generalized 2-soliton solution:
(13)u=−2k1φ1eθ1+k2φ2eθ2+a0k1+k2φ1φ2eθ1+θ2py,z+φ1eθ1+φ2eθ2+a0φ1φ2eθ1+θ2+γt,y,z,which is equal to Equation (6).ForN=3, based on the result of the generalized 1-soliton and 2-soliton solutions above and a=k1−k2/k1+k22, we should let
(14)θ3=k3x−k33t,aij=ki−kjki+kj2,1≤i<j≤3,F3=py,z+φ1eθ1+φ2eθ2+φ3eθ3+a12py,zφ1φ2eθ1+θ2+a13py,zφ1φ3eθ1+θ3+a23py,zφ2φ3eθ2+θ3+d123φ1φ2φ3eθ1+θ2+θ3and suppose that Equation (4) has solution of the following form:
(15)u=−2∂F3/∂xF3+γt,y,z,where d123 is to be determined. Substituting (15) into Equation (4), we have
(16)d123=a12a13a23py,z2.This implies that the function in (15) N=3 is a generalized 3-soliton solution of Equation (4). For N>3, we can obtain the N-soliton solution with the parallel manner.In conclusion, forN≥1, arbitrarily given constants ki and arbitrarily given differentiable functions gt,y, ht,z, ey,z, py,z, and φi=φiy,zi=1,2,⋯,N, let
(17)γ=t,y,z=gt,y+ht,z+ey,z,θi=kix−ki3ti=1,2,⋯,N,aij=ki−kjki+kj21≤i<j≤N,f=py,z+∑i=1Nfi,f1=∑i=1Nφ1expθi,fk=py,z1−k∑i1<i2<⋯<ikN∏j=1kφij∏im<inkaiminexp∑j=1kθij,1<k<N,⋯,fN=py,z1−N∏i=1Nφi∏i<jNaijexp∑i=1Nθi,where the notation ∑i1<i2<⋯<ikN means the summation over all possible combinations of 0<i1<i2<⋯<ik≤N,1<k<N. Then Equation (4) has the following generalized N-soliton solutions:
(18)u=−2∂f/∂xf+γt,y,z.Remark 1.
The N-soliton solutions N>2 contain arbitrarily differentiable function py,z, and this phenomenon has not been found in previous works. Because the N-soliton solutions we gained include a lot of arbitrarily differentiable functions, they have some interesting dynamic properties. We take 1- and 2-solitons to display some in Figures1 and 2.Remark 2.
Since these solutions contain arbitrary functions, we add the word “generalized” to each type of solutions.Figure 1
The graph of generalized 1-soliton solution for Equation (4), where x=t=0, k1=1, py,z=cos2ycos2z, φ1y,z=expysin2z, and γt,y,z=sinysinz. (a) Perspective view of the u in the y,z,u space and (b) overhead view of the u.
(a)
(b)Figure 2
The graph of generalized 2-soliton solution for Eq.(4), where x=t=0, k1=2, k2=1, py,z=cos2ycos2z, φ1y,z=expysin2z, φ2=y,z=expzsin2y, and γt,y,z=sinysinz. (a) Perspective view of the u in the y,z,u space and (b) overhead view of the u.
(a)
(b)


## 3. Conclusion

In this paper, we succeed in presenting generalizedN-soliton solutions by virtue of the special construction of Equation (4). We have obtained many new expressions of the solutions which are listed in Section 2. It is interesting that these expressions contain some arbitrary functions. Moreover, we can see that these generalized N-soliton solutions possess the same phase shift, besides the same dispersion relations. Furthermore, we are certain that this phenomenon will be found in other equation with similar construction and it will be our next goal to utilize our results to other physical equations.



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