Solitonic asymptotics for the Korteweg-de Vries equation in the small dispersion limit

TL;DR

This paper analyzes the small dispersion limit of the KdV equation near the trailing edge of oscillatory regions, revealing that oscillations form sharp soliton-like pulses through a Riemann-Hilbert asymptotic analysis.

Contribution

It provides a detailed asymptotic expansion in a critical scaling regime, showing how oscillations degenerate into soliton-like pulses near the trailing edge.

Findings

Oscillations become sharp pulses near the trailing edge.

Pulses resemble soliton solutions of KdV.

Asymptotic expansion obtained via Riemann-Hilbert method.

Abstract

We study the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ in a critical scaling regime where $x$ approaches the trailing edge of the region where the KdV solution shows oscillatory behavior. Using the Riemann-Hilbert approach, we obtain an asymptotic expansion for the KdV solution in a double scaling limit, which shows that the oscillations degenerate to sharp pulses near the trailing edge. Locally those pulses resemble soliton solutions of the KdV equation.