Abstract
In this paper, we discuss the existence of traveling wave solutions for compressible fluid equations by applying the theory and method of planar dynamical system, and obtain explicit expressions for all bounded traveling wave solutions by undetermined coefficient method, including kink and bell profile traveling wave solutions, as well as periodic wave solutions. We prove the kink profile solitary wave solution, both sides of which asymptotic values are not zero, is orbitally stable by the theory of Grillakis-Shatah-Strauss orbital stability.
Highlights
In this paper, we study traveling wave solutions and the orbital stability for the following compressible fluid equations with capillarity term υt − ux = 0, ut + p(υ)x + δυxxx = 0, (1.1)where x is the Lagrangian space variable, υ is specific volume, u is velocity, ε > 0 is the viscous coefficient, δ > 0 is the capillarity coefficient, and p(υ) is Van del Waals pressure
We discuss the existence of traveling wave solutions for compressible fluid equations by applying the theory and method of planar dynamical system, and obtain explicit expressions for all bounded traveling wave solutions by undetermined coefficient method, including kink and bell profile traveling wave solutions, as well as periodic wave solutions
We prove the kink profile solitary wave solution, both sides of which asymptotic values are not zero, is orbitally stable by the theory of GrillakisShatah-Strauss orbital stability
Summary
We study traveling wave solutions and the orbital stability for the following compressible fluid equations with capillarity term υt − ux = 0, ut + p(υ)x + δυxxx = 0,. Capillarity term δυxxx, called dispersive term, was firstly proposed by Korteweg,[8] and analyzed by Felderhor[9] and Bongonor.[10] δυxxx plays a role in restoring force in Eqs.(1.1) This term working with Van de Waals pressure leads to the solitary wave solution occurring. Shatah, and Strauss[16,17] abstracted and summarised the results from the previous references discussing orbital stability to solitary waves, and formed the theory called the theory of Grillakis-Shatah-Strauss orbital stability Based on this theory, Guo[18] studied the orbital stability of solitary wave solutions for nonlinear derivative Schörding equation; Zhang[19] discussed that for Kundun equation, while his result is more general than that in Ref. 18.
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