A note on decay rates of solutions to a system of cubic nonlinear Schr\"odinger equations in one space dimension

TL;DR

This paper investigates the decay rates of solutions to a system of cubic nonlinear Schrödinger equations with different masses in one dimension, establishing global existence and specific decay behavior under certain conditions.

Contribution

It provides new decay rate results for solutions of coupled cubic nonlinear Schrödinger equations with mass relations, under structural conditions on the nonlinearity.

Findings

Solutions exist globally for small initial data.

Decay rate of solutions is $O(t^{-1/2}(\log t)^{-1/2})$ in $L^\infty$.

Decay behavior depends on mass relations and nonlinearity structure.

Abstract

We consider the initial value problem for a system of cubic nonlinear Schr\"odinger equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution exists globally and decays of the rate $O(t^{-1/2}(\log t)^{-1/2})$ in $L^\infty$ as $t$ tends to infinity, if the system satisfies certain mass relations.