# Bifurcation Analysis and Single Traveling Wave Solutions of the Variable-Coefficient Davey–Stewartson System

Authors: Tianyong Han; Jiajin Wen; Zhao Li
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This paper mainly studies the bifurcation and single traveling wave solutions of the variable-coefficient Davey–Stewartson system. By employing the traveling wave transformation, the variable-coefficient Davey–Stewartson system is reduced to two-dimensional nonlinear ordinary differential equations. On the one hand, we use the bifurcation theory of planar dynamical systems to draw the phase diagram of the variable-coefficient Davey–Stewartson system. On the other hand, we use the polynomial complete discriminant method to obtain the exact traveling wave solution of the variable-coefficient Davey–Stewartson system.


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## Body

## 1. Introduction

Partial differential equations (PDEs) play a major role in the fields of plasma, quantum mechanics, and engineering [1]. In the study of PDEs, the most important thing is to analyze the dynamic behavior and find the exact traveling wave solution. In recent years, the study of exact traveling wave solutions of nonlinear PDEs with the variable coefficients has always been the focus of mathematicians and physicists, and many experts and scholars [2, 3] have proposed many methods to find PDEs with the variable coefficients, such as variable-coefficient extended mapping method [4], Hirota’s bilinear method [5], Lax integrability [6], and dynamical system approach [7, 8].One of the most important PDEs is the variable-coefficient Davey–Stewartson system. In this paper, we consider the variable-coefficient Davey–Stewartson system [9, 10]:(1)i∂u∂t+a1t∂2u∂x2+a2t∂2u∂y2−b1tu2u+b2tuv=0,s1∂2v∂x2−∂2v∂y2=s2∂2u2∂x2,where u=ut,x,y is the complex wave envelope, v=vt,x,y is the real forcing function, a1t and a2t are real functions with respect to time t, which stand for group velocity dispersion terms, b1t and b2t represent the quadratic nonlinearity and cubic nonlinear coefficient term, respectively, and s1 and s2 are constants. When the coefficients in equation (1) are constant, equation (1) is called the Davey–Stewartson system [11–15]. It is a very important nonlinear Schrödinger equation, which is usually used to describe the nonlinear wave packet of finite depth.In [10], Wei and her collaborators investigated equation (1) by the Lie group method and obtained the periodic solutions and elliptic function solutions. In [9], El-Shiekh and Gaballah obtained the dark soliton solutions and bright soliton solutions of equation (1) by the modified sine-Gordon equation method. Although some exact solutions of equation (1) have been obtained in references [9, 10], the analysis of the dynamic behavior and the classification of traveling wave solutions of this kind of equation have not been reported. Therefore, in this paper, we will further study the above two problems.The structure of this paper is as follows. In Section2, we use the bifurcation theory of planar dynamical systems to draw the phase diagram of the variable-coefficient Davey–Stewartson system. In Section 3, we construct the classification of all single traveling wave solutions of the variable-coefficient Davey–Stewartson system by the complete discrimination system. In Section 4, we give a summary.


## 2. Bifurcation Analysis of System (1)

Consider the traveling wave transformation as follows:(2)ut,x,y=Uξeiη,vt,x,y=Vξ,ξ=μ1x+μ2y−λt,η=k1x+k2y−θt,where μ1, μ2, k1, and k2 are constants and λt and θt are functions defined on t.Substituting (2) into (1) and separating the real and imaginary parts, we obtain(3)a1tμ12+a2tμ22Uξ+θ′t−a1tk12−a2tk22U−b1tU3+b2tUV=0,λ′t−2k1μ1a1t−2k2μ2a2tU′=0,s1μ12−μ22V″=μ12s2U2ξ″.In order to eliminate the termsU′ξ, we set λt=2∫μ1k1a1t+μ2k2a2tdt, θt=∫k12a1t+k22a2tdt+θ0, b1t=b1μ12a1t+μ22a2t, and b2t=b2μ12a1t+μ22a2t, where b1, b2, and θ0 are arbitrary constants. Then, equation (3) can be reduced to(4)U″ξ−b1U3ξ+b2UξVξ=0,s1μ12−μ22V″=μ12s2U2ξ″.Integrating the second equation of (4) twice with respect to ξ, we have(5)Vξ=s2μ12s1μ12−μ22U2ξ+c1ξ+c0,where c0 and c1 are integration constants. Substituting equation (5) into the first equation of (4), we obtain(6)U″ξ+s2μ12b2s1μ12−μ22−b1U3ξ+b2c0Uξ=0.Next, by assumingdU/dξ=ψ, we can rewrite equation (6) as the following planar dynamical system:(7)dUdξ=ψ,dψdξ=−s2μ12b2s1μ12−μ22−b1U3ξ−b2c0Uξ,with the Hamiltonian system(8)HU,ψ=12ψ2+14s2μ12b2s1μ12−μ22−b1U4ξ+b2c02U2ξ=h.Here, we assume thatGU=−s2μ12b2/s1μ12−μ22−b1U3ξ−b2c0Uξ. When s2μ12b2/s1μ12−μ22−b1b2c0<0, we easily obtain three zeros of GU including U0=0, U1=−b2c0s1μ12−μ22/s2μ12b2−b1s1μ12−μ22, and U2=−−b2c0s1μ12−μ22/s2μ12b2−b1s1μ12−μ22. When s2μ12b2/s1μ12−μ22−b1b2c0>0, we obtain U3=0. Then, we suppose that MiUi,0 are the equilibrium points of equation (8), and we obtain that MiUi,0 is the saddle point when G′Ui>0; MiUi,0 is the degraded saddle point when G′Ui=0; MiUi,0 is the center point when G′Ui<0. With the help of Maple software, we draw the phase portraits of (8) as shown in Figure 1.Figure 1
Phase portraits of system (7). (a)s2μ12b2/s1μ12−μ22−b1>0, b2c0/2<0. (b)s2μ12b2/s1μ12−μ22−b1<0, b2c0/2>0.
(a)(b)


## 3. Traveling Wave Solutions of System (1)

The complete discriminant system method was first introduced by Lu and his collaborators in 1996 [16]. In recent years, many experts and scholars [17–22] have applied this method to construct the exact traveling wave solutions of partial differential equations. In this section, we intend to use this method to analyze the exact traveling wave solution of the variable-coefficient Davey–Stewartson system.In fact, in Section2, we have simplified equation (1) to nonlinear equation (6). Next, multiply both ends of equation (6) by U′ and integrate once, and we obtain(9)U′2=d4U4ξ+d2U2ξ+d0,where d4=−1/2s2μ12b2/s1μ12−μ22−b1, d2=−b2c0, and d0 is the integration constant.Consider the following transformations:(10)U=±4d4−1/3w,p=4d24d4−2/3,q=4d04d4−1/3,ξ1=4d41/3ξ.Then, equation (9) can be rewritten as(11)wξ12=ww2+pw+q.Integrating equation (11) once, we obtain(12)±ξ1−ξ0=∫dwww2+pw+q,where ξ0 is an integration constant. Setting Fw=w2+pw+q, its complete discrimination system is(13)Δ=p2−4q.According to the root of equation (13), the traveling wave solution of equation (1) has four cases.Case 1.
Assume thatΔ=0. Since w>0, we can obtain(14)±ξ1−ξ0=∫dwww+p/2.
Ifp<0, the explicit solution of equation (1) is(15)u1t,x,y=±−12d2d4−1tanh212−2d24d4−2/31/24d41/3ξ−ξ0eik1x+k2y−θt,u2t,x,y=±−12d2d4−1coth212−2d24d4−2/31/24d41/3ξ−ξ0eik1x+k2y−θt.
Ifp>0, the explicit solution of equation (1) is(16)u3t,x,y=12d2d4−1tan2122d24d4−2/31/24d41/3ξ−ξ0eik1x+k2y−θt.
Ifp=0, the explicit solution of equation (1) is(17)u4t,x,y=24d4−1/34d41/3ξ−ξ02eik1x+k2y−θt.Case 2.
Assume thatΔ>0 and q=0. Since w>−p, we can obtain(18)±ξ1−ξ0=∫dwww+p.
According to Case1, the solution of equation (18) is as follows.
Ifp>0, the explicit solution of equation (1) is(19)u5t,x,y=±12d2d4−1tanh2122d24d4−1/31/24d41/3ξ−ξ0−d2d4−1eik1x+k2y−θt,u6t,x,y=±12d2d4−1coth2122d24d4−1/31/24d41/3ξ−ξ0−d2d4−1eik1x+k2y−θt.
Ifp<0, the explicit solution of equation (1) is(20)u7t,x,y=−12d2d4−1tan2122d24d4−1/31/24d41/3ξ−ξ0−d2d4−1eik1x+k2y−θt.Case 3.
Assume thatΔ>0, q≠0, and α<β<γ, where one of α, β, and γ is zero, and the rest of them are the two roots of Fw. Since α<w<β, make the transformation w=α+β−αsin2φ. It can be obtained from equation (13) that(21)±ξ1−ξ0=2γ−α∫dφ1−m2sin2φ,where m2=β−α/γ−α.
According to equation (21) and the definition of Jacobian elliptic function sn, the explicit solution of equation (1) is(22)u8t,x,y=4d4−1/3α+β−αsn2γ−α24d41/3ξ−ξ0,meik1x+k2y−θt.
Make the transformationw=−βsin2φ+γ/cos2φ and substitute it into equation (12). Similarly, the explicit solution of equation (1) can be obtained:(23)u9t,x,y=4d4−1/3−βsn2γ−α/24d41/3ξ−ξ0,m+γcn2γ−α/24d41/3ξ−ξ0,meik1x+k2y−θt.Case 4.
Assume thatΔ<0. Since w>0, we can make the following transformation:(24)w=qtan2φ2.
Substituting equation (24) into equation (13), we obtain(25)±ξ1−ξ0=q−1/4∫dφ1−k2sin2φ,where m2=1/21−p/2q.
According to equation (25) and the definition of Jacobian elliptic function cn, we obtain(26)cnq1/4ξ1−ξ0,m=cosφ.
From equation (24), we have(27)cosφ=2qw+q−1.
Comparing equation (26) with equation (27), we can obtain the explicit solution of equation (1):(28)u10t,x,y=4d4−1/324d04d4−1/31+cn4d04d4−1/31/44d41/3ξ−ξ0,m−4d04d4−1/3eik1x+k2y−θt.Remark 1.
In this paper, we obtained one of the solutions (ut,x,y) of equation (1). Using relation (5), we can obtain another solution vt,x,y of equation (1).


## 4. Conclusion

In this paper, the bifurcation and single traveling wave solutions of the variable-coefficient Davey–Stewartson system have been investigated by employing the bifurcation theory of planar dynamical systems and the polynomial complete discriminant method. The phase portraits of the variable-coefficient Davey–Stewartson system are shown in Figure1. Moreover, a series of new single traveling wave solutions is obtained. Compared with the published literature [9], this study not only obtains the hyperbolic function solution, trigonometric function solution, and rational function solution but also obtains the Jacobi function solution. We believe that the study of the variable-coefficient Davey–Stewartson system in the paper will help mathematicians and physicists.



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