The Nonlinear Schrodinger equation with a potential in dimension 1

TL;DR

This paper analyzes the long-term behavior of small solutions to the cubic nonlinear Schrödinger equation with a localized potential in one dimension, revealing nonlinear phase corrections influenced by the scattering matrix.

Contribution

It introduces a novel analysis of the nonlinear spectral measure and employs distorted Fourier transform techniques to understand asymptotic behavior in this setting.

Findings

Solutions exhibit nonlinear phase corrections depending on the scattering matrix.

The approach combines nonlinear spectral analysis with stationary phase and multilinear estimates.

The results apply to generic, localized potentials without bound states.

Abstract

We consider the cubic nonlinear Schrodinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, and does not have bound states, we obtain the long time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform - the so-called Weyl-Kodaira-Titchmarsh theory -, a precise understanding of the "nonlinear spectral measure" associated to the equation, and nonlinear stationary phase arguments as well as multilinear estimates in this distorted setting.