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Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

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Abstract

We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of \({2\pi}\)-periodic traveling wave solutions continuing from the zero state. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like |x|1/2 near its crest, the cusped waves obtained here behave like |x log |x||. Although the linear operator involved in this equation can be easily represented in terms of an integral operator, it maps continuous functions out of the Hölder and Lipschitz scales of function spaces by introducing logarithmic singularities. Since the nonlinearity is also of higher order than that of the unidirectional Whitham equation, several parts of our proofs and results deviate from those of the corresponding unidirectional equation, with the analysis of the logarithmic singularity being the most subtle component. This paper is part of a longer research programme for understanding the interplay between nonlinearities and dispersion in the formation of large-amplitude waves and their singularities.

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Authors and Affiliations

  1. Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Mats Ehrnström

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Mathew A. Johnson

  3. Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN, 47803, USA

    Kyle M. Claassen

Authors
  1. Mats Ehrnström
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  2. Mathew A. Johnson
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  3. Kyle M. Claassen
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Corresponding author

Correspondence to Mathew A. Johnson.

Additional information

Communicated by N. Masmoudi

ME was supported by Grant Nos. 231668 and 250070 from the Research Council of Norway; MJ was supported by the National Science Foundation under Grants DMS-1614785 and DMS-1211183; KC was supported by the National Science Foundation under Grant DMS-1211183.

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Ehrnström, M., Johnson, M.A. & Claassen, K.M. Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model. Arch Rational Mech Anal 231, 1635–1673 (2019). https://doi.org/10.1007/s00205-018-1306-5

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  • DOI: https://doi.org/10.1007/s00205-018-1306-5

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