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

)is 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.


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]: where u � u(t, x, y) is the complex wave envelope, v � v(t, x, y) is the real forcing function, a 1 (t) and a 2 (t) are real functions with respect to time t, which stand for group velocity dispersion terms, b 1 (t) and b 2 (t) represent the quadratic nonlinearity and cubic nonlinear coefficient term, respectively, and s 1 and s 2 are constants. When the coefficients in equation (1) are constant, equation (1) is called the Davey-Stewartson system [11][12][13][14][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. erefore, in this paper, we will further study the above two problems. e structure of this paper is as follows. In Section 2, 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.

Bifurcation Analysis of System (1)
Consider the traveling wave transformation as follows: where μ 1 , μ 2 , k 1 , and k 2 are constants and λ(t) and θ(t) are functions defined on t.
Substituting (2) into (1) and separating the real and imaginary parts, we obtain In order to eliminate the terms U ′ (ξ), we set and θ 0 are arbitrary constants. en, equation (3) can be reduced to Integrating the second equation of (4) twice with respect to ξ, we have where c 0 and c 1 are integration constants. Substituting equation (5) into the first equation of (4), we obtain Next, by assuming (dU/dξ) � ψ, we can rewrite equation (6) as the following planar dynamical system: with the Hamiltonian system Here, we assume that

Traveling Wave Solutions of System (1)
e complete discriminant system method was first introduced by Lu and his collaborators in 1996 [16]. In recent years, many experts and scholars [17][18][19][20][21][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.
Consider the following transformations: 2 Discrete Dynamics in Nature and Society en, equation (9) can be rewritten as Integrating equation (11) once, we obtain where ξ 0 is an integration constant. Setting F(w) � w 2 + pw + q, its complete discrimination system is According to the root of equation (13), the traveling wave solution of equation (1) has four cases.
If p < 0, the explicit solution of equation (1) is If p > 0, the explicit solution of equation (1) is If p � 0, the explicit solution of equation (1) is Case 2. Assume that Δ > 0 and q � 0. Since w > − p, we can obtain According to Case 1, the solution of equation (18) is as follows.
According to equation (25) and the definition of Jacobian elliptic function cn, we obtain cn (q) (1/4) From equation (24), we have Comparing equation (26) with equation (27), we can obtain the explicit solution of equation (1) Remark 1. In this paper, we obtained one of the solutions (u(t, x, y)) of equation (1). Using relation (5), we can obtain another solution v(t, x, y) of equation (1).

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. e phase portraits of the variable-coefficient Davey-Stewartson system are shown in Figure 1. 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.