Singularities of solutions to nonlinear Schr\"odinger equations

TL;DR

This paper investigates the propagation of singularities in solutions to nonlinear Schrödinger equations using wave packet transforms, providing conditions for solutions to belong to specific Sobolev spaces in certain directions.

Contribution

It introduces a new sufficient condition for solutions to nonlinear Schrödinger equations to be in Sobolev spaces based on wave front set analysis.

Findings

Identifies conditions for Sobolev regularity of solutions

Analyzes wave front set propagation in nonlinear Schrödinger equations

Provides a framework for understanding solution singularities

Abstract

We study the wave front set of the solutions of the initial value problem for nonlinear Schr\"{o}dinger equations via wave packet transform. We give an sufficient condition which assures that the solutions is in Sobolev space of order s in a given direction at a given time.