Analysis of quasi-periodic waves of cubic nonlinear Schr{\"o}dinger equations

TL;DR

This paper investigates quasi-periodic standing wave solutions of the cubic nonlinear Schrödinger equations, establishing a theoretical correspondence in the defocusing case and introducing a novel numerical scheme for energy minimization.

Contribution

It provides a new numerical scheme that handles mass and momentum constraints simultaneously and explores the relationship between ODE invariants and PDE conserved quantities.

Findings

Established a diffeomorphic correspondence in the defocusing case.

Developed a gradient flow-based numerical scheme with discrete renormalization.

Numerical experiments confirm the profile solutions as energy minimizers.

Abstract

We study the quasi-periodic standing wave solutions of the focusing and defocusing cubic nonlinear Schr{\"o}dinger equations in dimension one. In the defocusing case, we establish a diffeomorphic correspondence between the invariants of the ordinary differential equation of the wave profiles and the conserved quantities of the evolution equation. We introduce a numerical scheme to compute the minimizers of the energy at fixed mass and momentum for both focusing and defocusing cases. The scheme is based on a gradient flow approach with discrete renormalization at each time step. The novelty of our scheme is that the renormalization step deals at the same time with the mass and the momentum constraints. In numerical experiments, we observe that a given solution of the profile ordinary differential equation is also a minimizer of the energy at corresponding mass and momentum.