The Matrix Nonlinear Schr\"{o}dinger Equation with a Potential

TL;DR

This paper investigates the long-time behavior of small solutions to the matrix nonlinear Schrödinger equation with potential, showing they scatter like solutions to the linear equation, using spectral theory and factorization techniques.

Contribution

It provides a comprehensive analysis of scattering for small solutions in the supercritical regime with general boundary conditions and point interactions.

Findings

Small solutions exhibit scattering behavior as time approaches infinity.

The approach combines spectral theory with a novel factorization method.

Results hold for both generic and exceptional potentials.

Abstract

This paper is devoted to the study of the large-time asymptotics of the small solutions to the matrix nonlinear Schr\"{o}dinger equation with a potential on the half-line and with general selfadjoint boundary condition, and on the line with a potential and a general point interaction, in the whole supercritical regime. We prove that the small solutions are scattering solutions that asymptotically in time, $t \to\pm\infty, $ behave as solutions to the associated linear matrix Schr\"odinger equation with the potential identically zero. The potential can be either generic or exceptional. Our approach is based on detailed results on the spectral and scattering theory for the associated linear matrix Schr\"{o}dinger equation with a potential, and in a factorization technique that allows us to control the large-time behaviour of the solutions in appropriate norms.