Perturbation at blow-up time of self-similar solutions for the modified Korteweg-de Vries equation

TL;DR

This paper proves the stability of self-similar blow-up solutions for the modified Korteweg-de Vries equation, showing that small perturbations lead to solutions that asymptotically resemble the original blow-up profile.

Contribution

It provides the first stability result for self-similar blow-up solutions of the modified KdV equation on the line.

Findings

Existence of a unique global solution close to the self-similar blow-up profile.

Demonstration of stability of the blow-up behavior under small perturbations.

Asymptotic behavior of solutions as time approaches zero.

Abstract

We prove a first stability result of self-similar blow-up for the modified KdV equation on the line. More precisely, given a self-similar solution and a sufficiently small regular profile, there is a unique global solution which behaves at t tends to 0 as the sum of the self-similar solution and the smooth perturbation.