Nonlinear Dynamics

, Volume 98, Issue 1, pp 691–702 | Cite as

Evolution of initial discontinuity for the defocusing complex modified KdV equation

  • Liang-Qian Kong
  • Lei Wang
  • Deng-Shan WangEmail author
  • Chao-Qing Dai
  • Xiao-Yong Wen
  • Ling Xu
Original paper


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.


Whitham 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.

Compliance with ethical standards

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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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Liang-Qian Kong
    • 1
  • Lei Wang
    • 1
  • Deng-Shan Wang
    • 2
    Email author
  • Chao-Qing Dai
    • 3
  • Xiao-Yong Wen
    • 2
  • Ling Xu
    • 2
  1. 1.Department of Mathematics and PhysicsNorth China Electric Power UniversityBeijingChina
  2. 2.School of ScienceBeijing Information Science and Technology UniversityBeijingChina
  3. 3.School of SciencesZhejiang Agriculture and Forestry UniversityHangzhouChina

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