A KAM algorithm for the resonant non--linear Schrödinger equation

Abstract

In this note we use the normal forms of the completely resonant non--linear Schrödinger equation on a torus (NLS) derived in previous work in order to produce, under a KAM algorithm, large families of stable and unstable quasi periodic solutions for the NLS in any number of independent frequencies.

References in corpus (5)

• Resonant dynamics for the quintic non linear Schrödinger equation
• Quasi-Töplitz functions in KAM theorem
• Normal Form for the Schrödinger equation with analytic non--linearities
• Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
• Quasi-periodic solutions of the Schrödinger equation with arbitrary algebraic nonlinearities

Cited by in corpus (24)

• Time quasi-periodic gravity water waves in finite depth
• KAM for autonomous quasi-linear perturbations of mKdV
• Quasi-periodic solutions for fully nonlinear forced reversible Schroedinger equations
• Reducibility of the Quantum Harmonic Oscillator in $d$-dimensions with Polynomial Time Dependent Perturbation
• Traveling quasi-periodic water waves with constant vorticity
• Time quasi-periodic vortex patches of Euler equation in the plane
• KAM for beating solutions of the quintic NLS
• Growth of Sobolev norms for the quintic NLS on ${\mathbb{T}}^2$
• Long time stability of small finite gap solutions of the cubic Nonlinear Schrödinger equation on ${\mathbb{T}}^2$
• Reducible quasi-periodic solutions for the Non Linear Schrödinger equation
• The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation
• Stable and unstable time quasi periodic solutions for a system of coupled NLS equations
• Quasi-Periodic Solutions For Nonlinear Klein-Gordon Equations
• An infinite dimensional KAM theorem with application to two dimensional completely resonant beam equation
• On the stability of periodic multi-solitons of the KdV equation
• A reducibility result for a class of linear wave equations on ${\mathbb{T}}^d$
• Growth of Sobolev norms for the analytic NLS on ${\mathbb{T}}^2$
• Periodic solutions of a semilinear Euler-Bernoulli beam equation with variable coefficients
• Infinite energy quasi-periodic solutions to nonlinear Schrödinger equations on $\mathbb{R}$
• A KAM Theorem for Higher Dimensional Reversible Nonlinear Schrödinger Equations
• Response solutions for wave equations with variable wave speed and periodic forcing
• A separation lemma on sub-lattices
• The Existence of full dimensional tori for d-dimensional Nonlinear Schr$\ddot{\mbox{O}}$dinger equation
• On the Kolmogorov theorem for some infinite-dimensional Hamiltonian systems of short range