The Small-Dispersion Limit of the Intermediate Long Wave Equation via Semiclassical Soliton Ensembles

TL;DR

This paper investigates the small dispersion limit of the ILW equation, showing how solutions approximate initial data and Burgers' equation solutions over time, using semiclassical soliton ensembles and inverse scattering analysis.

Contribution

It introduces a rigorous analysis of the ILW equation's small dispersion limit via semiclassical soliton ensembles, extending inverse scattering techniques.

Findings

Proves $L^2$-convergence of ILW solutions to initial data at t=0.

Shows ILW solutions approximate Burgers' equation before gradient catastrophe.

Develops a formal WKB analysis for the ILW scattering problem.

Abstract

We study the small dispersion limit of the intermediate long wave (ILW) equation, specifically on a class of well-behaved initial conditions $u_0$ where the number of solitons in the solution increases without bound. First, we conduct a formal WKB-style analysis on the ILW direct scattering problem, generating approximate eigenvalues and norming constants. We then use this to define a modified set of scattering data and rigorously analyze the associated inverse scattering problem. The main results include demonstrating $L^2$-convergence of the solution at $t = 0$ to the original initial condition $u_0$ and for $0 < t < t_\mathrm{c}$ to the associated solution of invicid Burgers' equation, where $t_\mathrm{c}$ is the time of gradient catastrophe.