Numerical study of blow-up mechanisms for Davey-Stewartson II systems

TL;DR

This paper numerically investigates blow-up mechanisms in the focusing Davey-Stewartson II equation, suggesting self-similar blow-up profiles and dynamics similar to Ozawa's solution rather than standard NLS equations.

Contribution

It provides the first detailed numerical analysis of blow-up in Davey-Stewartson II systems, proposing a conjecture on the self-similar nature and scaling of blow-up profiles.

Findings

Blow-up appears to be self-similar in all studied cases.

The blow-up profile is a dynamically rescaled lump.

Scaling behavior matches Ozawa's explicit solution rather than standard NLS blow-up.

Abstract

We present a detailed numerical study of various blow-up issues in the context of the focusing Davey-Stewartson II equation. To this end we study Gaussian initial data and perturbations of the lump and the explicit blow-up solution due to Ozawa. Based on the numerical results it is conjectured that the blow-up in all cases is self similar, and that the time dependent scaling is as in the Ozawa solution and not as in the stable blow-up of standard $L^{2}$ critical nonlinear Schr\"odinger equations. The blow-up profile is given by a dynamically rescaled lump.