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Trigonometric shock waves in the Kaup–Boussinesq system

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Abstract

We consider the modulationally stable version of the Kaup–Boussinesq system which models propagation of nonlinear waves in various physical situations. It is shown that the Whitham modulation equations for this model have a new type of solutions which describe trigonometric shock waves. In the Gurevich–Pitaevskii problem of evolution of an initial discontinuity, these solutions correspond to a nonzero wave excitation on one of the sides of the discontinuity. As a result, the trigonometric shock wave propagates along a rarefaction wave and we consider the problem of the analytical description of such an evolution. Our analytical results are confirmed by numerical calculations.

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Funding

This work was supported by a grant from Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS.”

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  1. Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia

    Sergey K. Ivanov & Anatoly M. Kamchatnov

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  1. Sergey K. Ivanov
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  2. Anatoly M. Kamchatnov
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All authors contributed equally to this work

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Correspondence to Sergey K. Ivanov.

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Ivanov, S.K., Kamchatnov, A.M. Trigonometric shock waves in the Kaup–Boussinesq system. Nonlinear Dyn 108, 2505–2512 (2022). https://doi.org/10.1007/s11071-022-07326-5

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