Wellposedness of NLS in Modulation Spaces

TL;DR

This paper establishes new local and global well-posedness results for the 1D cubic nonlinear Schrödinger equation within modulation spaces, leveraging interpolation, conserved quantities, and integrability.

Contribution

It introduces novel well-posedness results in modulation spaces, combining multilinear interpolation and integrability-based conserved quantities.

Findings

Local well-posedness via multilinear interpolation

Global well-posedness using conserved quantities

Persistence of regularity in solutions

Abstract

We prove new local and global well-posedness results for the cubic one-dimensional nonlinear Schr\"odinger equation in modulation spaces. Local results are obtained via multilinear interpolation. Global results are proven using conserved quantities based on the complete integrability of the equation, persistence of regularity, and by separating off the time evolution of finitely many Picard iterates.