Long-time asymptotic analysis of the Korteweg-de Vries equation via the dbar steepest descent method: The Soliton region

TL;DR

This paper advances the understanding of long-time behavior of Korteweg-de Vries solutions in the soliton region by applying an improved dbar steepest descent method to low-regularity initial data.

Contribution

It introduces an enhanced asymptotic analysis for the KdV equation in the soliton region using the dbar steepest descent method under minimal regularity assumptions.

Findings

Provides sharper long-time asymptotic estimates in the soliton region.

Extends analysis to initial data with only finite moments.

Demonstrates effectiveness of the dbar steepest descent method for low-regularity data.

Abstract

We address the problem of long-time asymptotics for the solutions of the Korteweg-de Vries equation under low regularity assumptions. We consider rapidly decreasing initial data admitting only a finite number of moments. For the so-called "soliton region", an improved asymptotic estimate is provided, in comparison with the one already present in the literature. Our analysis is based on the dbar steepest descent method proposed by P. Miller and K. T. D. -R. McLaughlin.