Local well-posedness in weighted Sobolev spaces for nonlinear dispersive equations with applications to dispersive blow up

TL;DR

This paper establishes local well-posedness of nonlinear dispersive equations in weighted Sobolev spaces and explores dispersive blow-up phenomena for specific models like the Kawahara and Hirota-Satsuma equations.

Contribution

It introduces new well-posedness results in weighted Sobolev spaces and applies them to analyze dispersive blow-up in several nonlinear dispersive models.

Findings

Proved local well-posedness in weighted Sobolev spaces for various dispersive equations.

Identified conditions leading to dispersive blow-up in specific models.

Extended understanding of blow-up phenomena in nonlinear dispersive PDEs.

Abstract

In the first part of this work we study the local well-posedness of dispersive equations in the weighted spaces $H^s(\mathbb{R})\cap L^2(|x|^{2b}dx)$. We then apply our results for several dispersive models such as the Hirota-Satsuma system, the OST equation, the Kawahara equation and a fifth-order model. Using these local results, the second part of this work is devoted to obtain results related to dispersive blow up of the Kawahara equation and the Hirota-Satsuma system.