The final state problem for the nonlinear Schrodinger equation in dimensions 1, 2 and 3

TL;DR

This paper studies the long-term behavior of solutions to the defocusing nonlinear Schrödinger equation with time-dependent potentials in dimensions 1, 2, and 3, establishing convergence to a final state under certain conditions.

Contribution

It introduces a method to solve the large data final state problem for the nonlinear Schrödinger equation with metric perturbations using module regularity spaces.

Findings

Proves convergence of solutions to prescribed final states.

Handles nontrapping, compactly supported metric perturbations.

Extends analysis to higher dimensions with specific nonlinearities.

Abstract

In this article we consider the defocusing nonlinear Schr\"odinger equation, with time-dependent potential, in space dimensions $n=1, 2$ and $3$, with nonlinearity $|u|^{p-1} u$, $p$ an odd integer, satisfying $p \geq 5$ in dimension $1$, $p \geq 3$ in dimension $2$ and $p=3$ in dimension $3$. We also allow a metric perturbation, assumed to be compactly supported in spacetime, and nontrapping. We work with module regularity spaces, which are defined by regularity of order $k \geq 2$ under the action of certain vector fields generating symmetries of the free Schr\"odinger equation. We solve the large data final state problem, with final state in a module regularity space, and show convergence of the solution to the final state.