On scattering asymptotics for the 2D cubic resonant system

TL;DR

This paper proves scattering asymptotics for the 2D cubic resonant system, addressing unique challenges and establishing new estimates, which are crucial for understanding long-term behavior in related nonlinear Schrödinger equations.

Contribution

It introduces novel techniques to handle the 2D case's complexities, including weaker estimates and symmetry exploitation, advancing the analysis of resonant systems.

Findings

Proved scattering asymptotics for the 2D cubic resonant system.

Identified and addressed the failure of `$l^2$-estimate' in 2D.

Extended understanding of long-time dynamics in NLS on waveguides.

Abstract

In this paper, we prove scattering asymptotics for the 2D (discrete dimension) cubic resonant system. This scattering result was used in Zhao \cite{Z1} as an assumption to obtain the scattering for cubic NLS on $\mathbb{R}^2\times \mathbb{T}^2$ in $H^1$ space. Moreover, the 1D analogue is proved in Yang-Zhao \cite{YZ}. Though the scheme is also tightly based on Dodson \cite{D}, the 2D case is more complicated which causes some new difficulties. One obstacle is the failure of `$l^2$-estimate' for the cubic resonances in 2D (we also discuss it in this paper, which may have its own interests). To fix this problem, we establish weaker estimates and exploit the symmetries of the resonant system to modify the proof of \cite{YZ}. At last, we make a few remarks on the research line of `long time dynamics for NLS on waveguides'.