Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation

We prove a first stability result of self-similar blow-up for the modified Korteweg–de Vries 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 \(t=0\) $t=0$ as the sum of the self-similar solution and the smooth perturbation.