KAM for beating solutions of the quintic NLS

Abstract

We consider the nonlinear Schrödinger equation of degree five on the circle ${\mathbb{S}}^1=\mathbb{R}/2\pi$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.

32 pages, 1 figure

References in corpus (2)

• Growth of Sobolev norms for the quintic NLS on ${\mathbb{T}}^2$
• Reducible quasi-periodic solutions for the Non Linear Schrödinger equation

Cited by in corpus (9)

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• Stable and unstable time quasi periodic solutions for a system of coupled NLS equations
• Chaotic-like transfers of energy in Hamiltonian PDEs
• KAM for quasi-linear autonomous NLS
• Exponential Stability Estimates for the 1D NLS
• Periodic Solutions to Nonlinear Euler-Bernoulli Beam Equations
• Long time behavior of the NLS-Szeg{ö} equation
• Response solutions for wave equations with variable wave speed and periodic forcing