Long-time dynamics of small solutions to the Manakov system with initial data in the weighted $L^{2}$ space

TL;DR

This paper analyzes the long-time behavior of small solutions to the Manakov system with initial data in weighted L^2 space, advancing understanding of asymptotics in coupled nonlinear Schrödinger equations.

Contribution

It provides the first detailed long-time asymptotic analysis of the Manakov system for low regularity initial data using space-time resonance methods.

Findings

Derived asymptotic formulas for small solutions

Explored interaction between linear and modified scattering

Set groundwork for higher order AKNS systems analysis

Abstract

In this paper, we compute the long-time asymptotics for small solutions of the Manakov system which is a coupled system of nonlinear Schr\"odinger equations just under the assumption that the initial data lies in the weighted $L^{2}$ space. This will be our first step to understand the long-time asymptotics of higher order AKNS systems in low regularity spaces and analyze the interaction of modified scatterings. In the last section, we also discuss on the interaction of the linear and the modified scattering. Our techniques are relied on the idea of the space-time resonance.