Jerry Wang, Department of Applied Mathematics, University of Colorado Boulder

Modulation Evolution Along Multi Soliton Solutions of the KP II Equation

The Kadomtsev-Petviashvili II (KP II) equation is a partial differential equation that describes the evolution of weakly two-dimensional shallow water waves. The KP II Equation admits a variety of soliton solutions that represent localized nonlinear waves. In addition to line soliton solutions, KP II also admits more complex multi-"legged" soliton structures, such as Y- and X-shaped patterns.  Specifically, the evolution of long wavelength perturbations of 2-soliton solutions of KP II are investigated using modulation theory.  An initial-boundary value problem for a system of hyperbolic conservation laws is formulated for small and large amplitude perturbations to outgoing soliton legs, with this talk focusing on the resonant (Y-shaped) KP traveling wave solutions. The vertex where individuated soliton legs coalesce is modeled as a free boundary where information between legs can pass. The linearized hyperbolic system is solved exactly.  All admissible simple (rarefaction) waves of the nonlinear system are characterized. It is noted that perturbations of Y-shaped waves can lead to spontaneous leg deletion (the appearance of vacuum) or leg creation (gradient catastrophe) after passing through the vertex.