Modulation theory for soliton resonance and Mach reflection
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.
Cite
@article{arxiv.2110.13789,
title = {Modulation theory for soliton resonance and Mach reflection},
author = {Samuel J. Ryskamp and Mark A. Hoefer and Gino Biondini},
journal= {arXiv preprint arXiv:2110.13789},
year = {2022}
}