Near soliton dynamics and singularity formation for $L^2$ critical problems

TL;DR

This survey reviews recent advances in understanding singularity formation and soliton dynamics in mass critical dispersive equations like the nonlinear Schrödinger and generalized KdV equations.

Contribution

It provides a comprehensive overview of the classification of flow near solitons for these critical dispersive problems.

Findings

Classification of flow near solitons in critical regimes

Insights into singularity formation mechanisms

Summary of current research frontiers

Abstract

This survey reviews the state of the art concerning the singularity formation for two canonical dispersive problems: the mass critical non linear Schr\"odinger equation and the mass critical generalized KdV equation. In particular, we address the question of the classification of the flow for initial data near the soliton.