Solitary Waves and Periodic Waves in a Perturbed KdV Equation
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Solitary Waves and Periodic Waves in a Perturbed KdV Equation Hong Li1 · Hongquan Sun1 · Wenjing Zhu2 Received: 4 April 2020 / Accepted: 21 August 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we consider a perturbed Korteweg–de Vries (KdV) equation with weak dissipation and Marangoni effects. Main attention is focused on the existence conditions of periodic and solitary wave solutions of the perturbed KdV equation. Based on bifurcation theory of dynamic system and geometric singular perturbation method, the parameter conditions and wave speed conditions for the existence of one periodic solution, one solitary solution and the coexistence of a solitary solution and infinite number of periodic solutions are given. By using Chebyshev criterion to analyze the ratio of Abelian integrals, the monotonicity of wave speed is proved, and the upper and lower bounds of wave speed are obtained. Keywords KdV equation · Traveling wave · Chebyshev system · Abelian integral
1 Introduction Traveling wave solutions are an important type of solutions for nonlinear wave equations. Many nonlinear wave equations have many kinds of traveling wave solutions. But the traveling wave solution is very sensitive to the external influence [4]. In practice, there are often small external perturbations. In order to simulate these small perturbations, the perturbed terms are often added to the wave equation to generate the perturbed nonlinear wave equation. For a long time, because of the remarkable physical properties, KdV equation has attracted wide attention in stratified internal wave, ion acoustic wave, plasma physics and so on [1,14]. The exact solitary wave and periodic solutions of KdV equation have been studied extensively [2,11,13,17]. Recently, some researchers have paid attention
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Hong Li [email protected]
1
Department of Mathematics, Jiujiang University, Jiujiang 332005, People’s Republic of China
2
Department of Mathematics, China Jiliang University, Hangzhou 310018, People’s Republic of China 0123456789().: V,-vol
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to the perturbed KdV equation. For example, Ogawa [20] studied the existence of solitary wave solutions and periodic wave solutions of the perturbed KdV equation u t + uu x + u x x x + ε(u x x + u x x x x ) = 0
(1.1)
Ogawa solved two problems of Eq. (1.1). One is the persistence of solitary and periodic waves, and the other is the selection principle of wave speed. For 0 < ε 1, Ogawa indicated that the system (1.1) has periodic solutions when the wave speed is in a certain range, and there is a unique solitary solution at a certain speed. Zhuang and Du [25] studied a singularly perturbed higher-order KdV equation u t + αu n+1 u x + βu x x x + ε(u x x + u x x x x ) = 0.
(1.2)
It was proved that the solitary wave solution of the equation persist when the perturbation parameters are small enough. On the interphase boundary accompanied with heat or mass transfer processes, the inhomogeneous distribution of temperature or solute concentration (or inhomogeneous
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