Scattering problem for the generalized Korteweg-de Vries equation

TL;DR

This paper investigates the scattering behavior of solutions to the generalized Korteweg-de Vries equation, establishing small data scattering in weighted Sobolev spaces and providing new characterizations of scattering phenomena.

Contribution

It introduces two equivalent characterizations of scattering in weighted Sobolev spaces and incorporates Fourier-Lebesgue space criteria into the analysis.

Findings

Proves small data scattering in weighted Sobolev spaces.

Provides new characterizations of scattering phenomena.

Integrates Fourier-Lebesgue space criteria into scattering analysis.

Abstract

In this paper we study the scattering problem for the initial value problem of the generalized Korteweg-de Vries (gKdV) equation. The purpose of this paper is to achieve two primary goals. Firstly, we show small data scattering for (gKdV) in the weighted Sobolev space, ensuring the initial and the asymptotic states belong to the same class. Secondly, we introduce two equivalent characterizations of scattering in the weighted Sobolev space. In particular, this involves the so-called conditional scattering in the weighted Sobolev space. A key ingredient is incorporation of the scattering criterion for (gKdV) in the Fourier-Lebesgue space by the authors into the the scattering problem in the weighted Sobolev space.