Effective Dynamics of the Vector Nonlinear Schr\"odinger Equations on Large Domains

TL;DR

This paper derives an effective resonant equation for the vector nonlinear Schrödinger equation on large periodic domains, analyzing its properties and long-time dynamics for small initial data.

Contribution

It introduces the continuous resonant equation as an effective model for large domains and small data, and studies its Hamiltonian structure and well-posedness.

Findings

Derivation of the continuous resonant equation for large domains

Analysis of Hamiltonian structure of the resonant equation

Establishment of well-posedness and properties of the resonant dynamics

Abstract

We consider the vector nonlinear Schr\"odinger equation posed on the box with periodic boundary conditions, and derive the continuous resonant (CR) equation that describes the effective dynamics for large box size and small data size over very large time-scales. Moreover, we investigate various properties of the continuous resonant equations, including the Hamiltonian structure and the well-posedness, etc..