Scattering theory for the Schr\"odinger-Debye System

TL;DR

This paper investigates the long-term behavior of solutions to the Schr"odinger-Debye system across various dimensions, establishing global existence, scattering, and modified scattering results for small initial data.

Contribution

It provides the first comprehensive analysis of global existence and scattering for the Schr"odinger-Debye system in multiple dimensions, including a modified scattering result in one dimension.

Findings

Global existence and scattering for small solutions in dimensions 2, 3, 4.

Modified scattering result in dimension 1.

Results extend understanding of nonlinear Schr"odinger-type systems.

Abstract

We study the Schr\"odinger-Debye system over $\mathbb{R}^d$ iu_t+\frac 12\Delta u=uv,\quad \mu v_t+v=\lambda |u|^2 and establish the global existence and scattering of small solutions for initial data in several function spaces in dimensions $d=2,3,4$. Moreover, in dimension $d=1$, we prove a Hayashi-Naumkin modified scattering result.