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The Complete Classification of Solutions to the Riemann Problem of the Defocusing Complex Modified KdV Equation

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Abstract

The complete classification of solutions to the defocusing complex modified KdV equation with step-like initial condition is studied by the finite-gap integration approach and Whitham modulation theory. All kinds of combination solutions consisting of genus-0 regions, genus-1 regions, or genus-2 regions are found by classifying the Riemann invariants. The behaviors of wave breaking in Riemann problem of the defocusing complex modified KdV equation are much richer and more complicated than those in the nonlinear Schrödinger equation. It is demonstrated that a large oscillating region can be composed of four basic genus-1 dispersive shock waves, a case of solution may be consisted of up to six regions, and the plateau, vacuum, rarefaction wave, and dispersive shock wave can coexist in the same solution region. Moreover, the genus-2 region, produced from the collision of two dispersive shock waves, is described detailedly by the genus-2 Whitham equations. The direct numerical simulations on the defocusing complex modified KdV equation show remarkable agreement with the results from Whitham modulation theory.

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Acknowledgements

The authors gratefully thank the referees for their valuable suggestions. The work of DSW is supported by National Natural Science Foundation of China under Grant No. 11971067, the Fundamental Research Funds for the Central Universities under Grant No. 2020NTST22, and the Beijing Great Wall Talents Cultivation Program under Grant No. CIT&TCD20180325. The work of ZX is supported by Beijing outstanding talents training fund youth top individual project and Premium Funding Project for Academic Human Resources Development in Beijing Union University under Grant No. BPHR2020EZ01.

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Deng-Shan Wang

  2. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Ling Xu

  3. Institute of Fundamental and Interdisciplinary Sciences, Beijing Union University, Beijing, 100101, People’s Republic of China

    Zuxing Xuan

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  1. Deng-Shan Wang
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  2. Ling Xu
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  3. Zuxing Xuan
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Correspondence to Deng-Shan Wang.

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Communicated by Peter Miller.

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Wang, DS., Xu, L. & Xuan, Z. The Complete Classification of Solutions to the Riemann Problem of the Defocusing Complex Modified KdV Equation. J Nonlinear Sci 32, 3 (2022). https://doi.org/10.1007/s00332-021-09766-6

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