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Exact Solitary Water Waves with Vorticity

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Abstract

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA

    Vera Mikyoung Hur

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  1. Vera Mikyoung Hur
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Correspondence to Vera Mikyoung Hur.

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Communicated by J. M. Ball

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Hur, V.M. Exact Solitary Water Waves with Vorticity. Arch Rational Mech Anal 188, 213–244 (2008). https://doi.org/10.1007/s00205-007-0064-6

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  • DOI: https://doi.org/10.1007/s00205-007-0064-6

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