Published: Dec. 10, 2019

Justin Cole, Department of Applied Mathematics, University of Colorado Boulder

Soliton Dynamics in the Korteweg-de Vries Equation with Nonzero Boundary Conditions

Inspired by recent experiments, the Korteweg-de Vries equation with nonzero Dirichlet boundary conditions is considered. Two types of boundary data are examined: step up (which generates a rarefaction wave) and step down (which creates a dispersive shock wave). Soliton dynamics are analytically studied via inverse scattering transform and a small dispersion asymptotic approximation. Depending on it's initial position and amplitude, an incident soliton will either transmit through or become trapped inside a step-induced wave. A formula for the transmitted soliton and it's phase shift is derived. The asymptotic approximation provides a description of the trapped soliton dynamics. Finally, direct numerics are shown to agree well with the analytical results.