Lie symmetry analysis, bifurcations and exact solutions for the (2+1)-dimensional dissipative long wave system
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Lie symmetry analysis, bifurcations and exact solutions for the (2+1)-dimensional dissipative long wave system Lina Chang1 · Hanze Liu1 · Xiangpeng Xin1 Received: 24 September 2019 / Revised: 18 April 2020 / Accepted: 9 June 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract By the combination of Lie symmetry analysis and dynamical system method, the (2+1)-dimensional dissipative long wave system is studied. First, we get Lie algebra and Lie symmetry group of the system. Then, by using the dynamical system method, the bifurcation and phase portraits of the corresponding traveling system of the system are obtained, it is shown that for different parametric space, the system has infinitely many solitary wave solutions, periodic wave solutions, kink or anti kink wave solutions. At last, the conservation laws of the system are given. Keywords (2+1)-dimensional dissipative long wave system · Lie symmetry analysis · Bifurcation · Traveling wave solution · Conservation law Mathematics Subject Classification 37K10 · 35C05
1 Introduction In mathematics and physics, partial differential equations (PDEs) with nonlinear terms are called nonlinear partial differential equations (NLPDEs). In nature, nonlinear partial differential equations can describe many nonlinear phenomena, so finding some new methods to solve nonlinear partial differential equations is one of the important tasks of studying nonlinear science, which is helpful for people to better understand the physical significance of nonlinear phenomena. So far, a large number of scholars have proposed many effective methods to find exact solutions of nonlinear partial differential equations, for example, the sin–cos method [1], the Jacobi elliptic function method
This work was supported by the National Natural Science Foundation of China under Grant Nos. 11171041 and 11505090, the high-level personnel foundation of Liaocheng University under Grant Nos. 31805 and 318011613.
B 1
Hanze Liu [email protected] School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, Shandong, China
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L. Chang et al.
[2,3], the exp(−φ(ξ ))-expansion method [4,5], the ( GG )-expansion method [6,7], the Bäcklund transformation method [8], the alternative functional variable method [9], the Lie group method [10–14], the dynamical system [15–20], and so on. In this paper, we study the following (2+1)-dimensional dissipative long wave system: u t y + vx x + α(u 2 )x y = 0, vt + β(uv)x + u x x y = 0.
(1.1)
where u = u(x, y, t) and v = v(x, y, t) denote the unknown functions, α and β are nonzero constants. The (2+1)-dimensional dissipative long wave system is a famous nonlinear model in physical application, nonlinear wave theory and nonlinear science. Zeng solved the (2+1)-dimensional dispersive long wave equation by using a new modified algebraic method, and obtained abundant new exact solutions in [21]. Liu studied this system by the modified CK’s direct method in [22]. Zhang used the auxiliary equation and the expanded mapping m
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