Samuel Ryskamp, Department of Applied Mathematics, University of Colorado Boulder
Modulation theory for Miles resonance and Mach reflection
Mach reflection occurs when a sufficiently large amplitude line soliton interacts with a barrier at a sufficiently small angle. A Y-shaped resonant triad is then formed consisting of two smaller amplitude solitons and a larger “Mach” stem. In this talk, I will present a new description of these Y-shaped "Miles" solitons, traveling wave solutions to the Kadomtsev-Petviashvili II (KPII) equation, as discontinuous shock solutions to an infinite family of soliton modulation equations.
The fully two-dimensional soliton modulation equations, valid in the zero-dispersion limit of the KPII equation, are first shown 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 compresses to a travelling discontinuity, which is a multivalued, weak solution to the one-dimensional modulation equations. These results are then used to analytically describe the dynamics of the Mach reflection problem where a line soliton is incident upon an inward oblique corner, reformulated as a Riemann problem for the modulation equations. Modulation theory results show excellent agreement with direct KPII numerical simulation, and they also reveal a remarkable symmetry with the dynamics of a soliton incident upon an outward oblique corner (the Mach expansion problem). More broadly, this local description of Y-solitons within the context of modulation theory enables the construction of approximate solutions to more complex problems.