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arXiv · 2110.13789

Modulation theory for soliton resonance and Mach reflection

Samuel J. Ryskamp·Mark A. Hoefer·Gino Biondini
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Abstract

Resonant Y-shaped soliton solutions to the Kadomtsev-Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully two-dimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine-Hugoniot jump conditions for the one-dimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, V-shaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation.

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Samuel J. Ryskamp, Mark A. Hoefer, Gino Biondini. 2021-10-26. Modulation theory for soliton resonance and Mach reflection. https://doi.org/10.1098/rspa.2021.0823

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