Global dynamics of the $2d$ NLS with white noise potential and generic polynomial nonlinearity

TL;DR

This paper establishes a method to construct unique global solutions for the 2D nonlinear Schrödinger equation with white noise potential and polynomial nonlinearities, extending previous results to more general nonlinearities.

Contribution

It introduces a novel approach based on Hairer-Labbé's method to handle arbitrary polynomial nonlinearities in the 2D NLS with white noise potential, improving upon prior work limited to specific nonlinearities.

Findings

Successfully constructs global solutions as limits of smoothed potential solutions.

Extends previous results to more general polynomial nonlinearities.

Provides a framework for analyzing stochastic PDEs with irregular potentials.

Abstract

Using an approach introduced by Hairer-Labb\' e we construct a unique global dynamics for the NLS on $\T^2$ with a white noise potential and an arbitrary polynomial nonlinearity. We build the solutions as a limit of classical solutions (up to a phase shift) of the same equation with smoothed potentials. This is an improvement on previous contributions of us and Debussche-Weber dealing with quartic nonlinearities and cubic nonlinearities respectively.