# Self-similar dynamics for the modified Korteweg-de Vries equation

## TL;DR

This paper establishes local well-posedness for the modified Korteweg-de Vries equation in a critical space that includes self-similar solutions, enabling analysis of solution behavior near blow-up and at infinity.

## Contribution

It introduces a critical space framework for the mKdV equation that incorporates self-similar solutions, allowing detailed study of solution dynamics and blow-up behavior.

## Key findings

- Asymptotic description of small solutions as t approaches infinity
- Construction of solutions with prescribed blow-up behavior at t approaching zero
- Proof of local well-posedness in a self-similar critical space

## Abstract

We prove a local well posedness result for the modified Korteweg-de Vries equation in a critical space designed so that is contains self-similar solutions. As a consequence, we can study the flow of this equation around self-similar solutions: in particular, we give an as-ymptotic description of small solutions as t $\rightarrow$ +$\infty$ and construct solutions with a prescribed blow up behavior as t $\rightarrow$ 0.