The Bifurcations and Exact Traveling Wave Solutions for a Nonlocal Hydrodynamic-Type System
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The Bifurcations and Exact Traveling Wave Solutions for a Nonlocal Hydrodynamic-Type System Jianli Liang1,2
· Yi Zhang1
Received: 26 September 2019 / Revised: 12 June 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A hydrodynamic-type system taking into account nonlocal effects is investigated. The exact traveling wave solutions including smooth solitary waves solutions, pseudo-peakons, periodic peakons, compactons, kink, and anti-kink wave solutions and so on are derived via the method of dynamical systems and the theory of singular traveling wave systems. It is worth pointing out that the uncountably infinitely many solitary wave solutions, kink, and anti-kink solutions are new solutions for the modeling system. Keywords Singular traveling wave system · Bifurcations · Pseudo-peakon · Periodic peakon Mathematics Subject Classification (v) 34A05 · 35Q51 · 35Q53
1 Introduction Solitary wave theory is a rapidly developing area, and a large variety of sophisticated ideas and many powerful mathematical methods have been proposed [1–3]. Classification of solitary wave bifurcation and exact solitary wave solutions play an important role in understanding the dynamical behaviors of the solitary wave, and bifurcation classification and exact solutions of a vast number of nonlinear models have been studied [2, 4–12]. When the propagation constant of solitary waves or physical parameters in the nonlinear wave equations changes, bifurcations of solitary waves occur and different types of waves arise. Lenells proposed an approach to classify the bounded traveling waves [5], and presented a
Yi Zhang
[email protected] Jianli Liang [email protected] 1
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, People’s Republic of China
2
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, People’s Republic of China
Jianli Liang and Yi Zhang
qualitative analysis of the traveling wave solutions for some nonlinear wave equations [4– 6]. Yang classified the bifurcations of solitary wave to three major types named saddle-node, pitchfork, and transcritical bifurcations for a generalized nonlinear Schr¨odinger equation [11]. Furthermore, Li and his cooperators not only classified traveling wave solutions, but also obtained the expression of exact solutions effectively [2, 7, 8, 12]. With the development of solitary wave theory, a vast number of nonlinear models have emerged. In Ref. [13], Vladimirov et al. deal with some modeling systems of PDE, taking into account nonlocal effects. These effects are manifested when an intense pulse loading (impact, explosion, etc.) is applied to media, possessing an internal structure on mesoscale. In the long wave approximation, the main classical form of balance equations for mass and momentum of media in the one-dimensional case can be written as follows: ut + px = 0, ρt + ρ 2 ux = 0, where u is the mass velocity, p is the pressure, ρ is the density, t is the time, x is the mass (Lagrangian) coordinate. Th
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