Evolution of initial discontinuity for the defocusing complex modified KdV equation

• Published in: Nonlinear dynamics 2019-10, Vol.98 (1), p.691-702
• Main Authors: Kong, Liang-Qian, Wang, Lei, Wang, Deng-Shan, Dai, Chao-Qing, Wen, Xiao-Yong, Xu, Ling
• Format: Article
• Language: English
• Subjects: Automotive Engineering; Classical Mechanics; Classification; Computer simulation; Control; Defocusing; Dynamical Systems; Engineering; Evolution; Fluid mechanics; Korteweg-Devries equation; Linear equations; Mechanical Engineering; Original Paper; Rarefaction; Schrodinger equation; Shock waves; Vibration; Wave dispersion
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-019-05222-z

The complete classification of solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation with the step-like initial condition is given by Whitham theory. The process of studying the solution of cmKdV equation can be reduced to explore four quasi-linear equations, which predicts the evolution of dispersive shock wave. The results obtained here are quite different from the defocusing nonlinear Schrödinger equation: the bidirectionality of defocusing nonlinear Schrödinger equation determines that there are two basic rarefaction and shock structures while in the cmKdV case three basic rarefaction structures and four basic dispersive shock structures are constructed which lead to more complicated classification of step-like initial condition, and wave patterns even consisted of six different regions while each of wave patterns is consisted of five regions in the defocusing nonlinear Schrödinger equation. Direct numerical simulations of cmKdV equation are agreed well with the solutions corresponding to Whitham theory.