Evolution of initial discontinuity for the defocusing complex modified KdV equation
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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.
KeywordsWhitham equations Dispersive shock wave Complex mKdV equation
This work is supported by National Natural Science Foundation of China under Grant Nos. 11875126 and 11971067, Beijing Natural Science Foundation under Grant No. 1182009, the Beijing Great Wall Talents Cultivation Program under Grant No. CIT&TCD20180325 and Qin Xin Talents Cultivation Program (Nos. QXTCP A201702 and QXTCP B201704) of Beijing Information Science and Technology University.
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Conflict of Interest
We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the paper submitted.
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