Modulation theory for soliton resonance and Mach reflection

TL;DR

This paper develops a modulation theory framework for understanding Y-shaped soliton resonance and Mach reflection phenomena in the KPII equation, linking multi-dimensional modulation equations to observable shock and reflection behaviors.

Contribution

It introduces a reduction of two-dimensional modulation equations to a one-dimensional system and applies this to analytically describe Mach reflection dynamics.

Findings

Excellent agreement between modulation theory and numerical simulations.

Reduction of 2D modulation equations to 1D in the zero dispersion limit.

Analytical description of Mach reflection for V-shaped initial conditions.

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.