Global solution of the initial value problem for the focusing Davey-Stewartson II system

TL;DR

This paper establishes the existence of global solutions for the focusing Davey-Stewartson II system with large initial data, showing blow-up can only occur on a bounded real analytic variety, thus advancing understanding of singularity formation.

Contribution

It constructs global solutions for a dense set of initial data in $L^2$, without smallness assumptions, and characterizes the blow-up set for the focusing Davey-Stewartson II system.

Findings

Global solutions exist for a dense set of initial data in $L^2$.

Blow-up occurs only on a bounded real analytic variety.

Solutions may have singularities, but are globally defined outside the blow-up set.

Abstract

We consider the two dimensional focusing Davey-Stewartson II system and construct the global solution of the Cauchy problem for a dense in $L^2(\mathbb C)$ set of initial data. We do not assume that the initial data is small. So, the solutions may have singularities. We show that the blow-up may occur only on a real analytic variety and the variety is bounded in each strip $t \leq T$.