Interactions and Asymptotics of Dispersive Shock Waves -- Korteweg-de Vries Equation

TL;DR

This paper analyzes the long-time behavior of dispersive shock waves in the Korteweg-de Vries equation with step-like initial data, showing how multiple DSWs interact and merge into a single dominant wave over time.

Contribution

It provides a detailed asymptotic analysis of DSW interactions and demonstrates the merging process into a single-phase wave using inverse scattering and matched-asymptotic methods.

Findings

Multiple DSWs form from step-like initial data

Interacting DSWs merge into a single dominant wave

The behavior is analogous to viscous shock waves in Burgers' equation

Abstract

The long-time asymptotic solution of the Korteweg-de Vries equation for general, step-like initial data is analyzed. Each sub-step in well-separated, multi-step data forms its own single dispersive shock wave (DSW); at intermediate times these DSWs interact and develop multiphase dynamics. Using the inverse scattering transform and matched-asymptotic analysis it is shown that the DSWs merge to form a single-phase DSW, which is the `largest' one possible for the boundary data. This is similar to interacting viscous shock waves (VSW) that are modeled with Burgers' equation, where only the single, largest-possible VSW remains after a long time.