A direct construction of a full family of Whitham solitary waves

Mats Ehrnström; Katerina Nik; Christoph Walker

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arXiv · 2204.03274

A direct construction of a full family of Whitham solitary waves

Mats Ehrnström·Katerina Nik·Christoph Walker

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Abstract

Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed-wave height space, and reaches a singular highest wave at $\varphi(0) = \frac{\mu}{2}$. The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).

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Mats Ehrnström, Katerina Nik, Christoph Walker. 2022-04-07. A direct construction of a full family of Whitham solitary waves. https://arxiv.org/abs/2204.03274

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