Analysis of Quasi-Periodic Waves of Cubic Nonlinear Schrödinger Equations

Abstract

We study the quasi-periodic standing wave solutions of the focusing and defocusing cubic nonlinear Schrödinger 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, while existing schemes in the literature cannot handle more than one constraint. 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.

Keywords:

• nonlinear Schrödinger equation Keyword: nonlinear Schrödinger equation
• standing waves Keyword: standing waves
• normalized gradient flow Keyword: normalized gradient flow
• variational method Keyword: variational method
• quasi-periodic solutions Keyword: quasi-periodic solutions

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