Polynomial deceleration for a system of cubic nonlinear Schr\"{o}dinger equations in one space dimension

TL;DR

This paper investigates the long-term behavior of solutions to a specific cubic nonlinear Schr"{o}dinger system in one dimension, revealing a novel polynomial decay deceleration caused by nonlinear amplification effects.

Contribution

It is the first to demonstrate polynomial order decay deceleration in a nonlinear Schr"{o}dinger system, expanding understanding beyond previously known logarithmic decay results.

Findings

Solutions decay slower than linear solutions as t approaches infinity

Decay rate difference is of polynomial order due to nonlinear effects

First model showing polynomial decay deceleration in this context

Abstract

In this paper, we consider the initial value problem of a specific system of cubic nonlinear Schr\"{o}dinger equations. Our aim of this research is to specify the asymptotic profile of the solution in $L^{\infty}$ as $t \to \infty$. It is then revealed that the solution decays slower than a linear solution does. Further, the difference of the decay rate is a polynomial order. This deceleration of the decay is due to an amplification effect by the nonlinearity. This nonlinear amplification phenomena was previously known for several specific systems, however the deceleration of the decay in these results was by a logarithmic order. As far as we know, the system studied in this paper is the first model in that the deceleration in a polynomial order is justified.