Asymptotic behavior in time of solution to system of cubic nonlinear Schr"odinger equations in one space dimension

TL;DR

This paper investigates the long-term behavior of solutions to coupled cubic nonlinear Schrödinger equations in one dimension, revealing new oscillatory asymptotic profiles and classifying diverse solution behaviors.

Contribution

It introduces a novel asymptotic profile involving a sum of oscillating parts and extends classification results for cubic nonlinear Schrödinger systems.

Findings

Existence of solutions with asymptotic profiles as sums of oscillating components

Identification of solution behaviors such as modified scattering, amplification, and dissipation

Extension of previous classification results for cubic nonlinear Schrödinger systems

Abstract

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Schr"odinger equations in one space dimension. It turns out that for a system there exists a small solution of which asymptotic profile is a sum of two parts oscillating in a different way. This kind of behavior seems new. Further, several examples of systems which admit solution with several types of behavior such as modified scattering, nonlinear amplification, and nonlinear dissipation, are given. We also extend our previous classification result of nonlinear cubic systems.