Resonant dynamics for the quintic nonlinear Schrödinger equation

DOI10.1016/j.anihpc.2012.01.005zbMath1259.37045arXiv1111.6827OpenAlexW2152283434MaRDI QIDQ432273

Benoît Grébert, Laurent Thomann

Publication date: 4 July 2012

Published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1111.6827

zbMATH Keywords

normal formsenergy exchangenonlinear Schrodinger equation

Mathematics Subject Classification ID

Stability in context of PDEs (35B35) NLS equations (nonlinear Schrödinger equations) (35Q55) Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems (37K45) Resonance in context of PDEs (35B34)

Related Items (29)

Birkhoff normal forms for Hamiltonian PDEs in their energy space ⋮ Resonances and genericity in Birkhoff normal forms ⋮ Sobolev norms explosion for the cubic NLS on irrational tori ⋮ Growth of Sobolev norms for the analytic NLS on \(\mathbb{T}^2\) ⋮ Global existence and long-term behavior of a nonlinear Schrödinger-type equation ⋮ On the quasi-periodic solutions for generalized Boussinesq equation with higher order nonlinearity ⋮ KAM tori for defocusing modified KDV equation ⋮ An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS ⋮ Dynamics of nonlinear Klein-Gordon equations in low regularity on \(\mathbb{S}^2\) ⋮ Normal form and dynamics of the Kirchhoff equation ⋮ Energy transfer model and large periodic boundary value problem for the quintic nonlinear Schrödinger equations ⋮ Bose-Einstein condensation under the cubic-quintic Gross-Pitaevskii equation in radial domains ⋮ Stable and unstable time quasi periodic solutions for a system of coupled NLS equations ⋮ Modified scattering and beating effect for coupled Schrödinger systems on product spaces with small initial data ⋮ Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces ⋮ A KAM algorithm for the resonant non-linear Schrödinger equation ⋮ Transfers of energy through fast diffusion channels in some resonant PDEs on the circle ⋮ Energy transfer model for the derivative nonlinear Schrödinger equations on the torus ⋮ KAM for beating solutions of the quintic NLS ⋮ Chaotic-like transfers of energy in Hamiltonian PDEs ⋮ Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs ⋮ Rational normal forms and stability of small solutions to nonlinear Schrödinger equations ⋮ Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations ⋮ On the normal form of the Kirchhoff equation ⋮ Exponential and sub-exponential stability times for the NLS on the circle ⋮ Construction of quasi-periodic solutions for the quintic Schrödinger equation on the two-dimensional torus $\mathbb {T}^2$ ⋮ KAM tori for the two-dimensional completely resonant Schrödinger equation with the general nonlinearity ⋮ Stroboscopic averaging for the nonlinear Schrödinger equation ⋮ On Scattering for the Quintic Defocusing Nonlinear Schrödinger Equation on R × T²

Cites Work

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• On the energy exchange between resonant modes in nonlinear Schrödinger equations
• Birkhoff coordinates for the focusing NLS equation
• Birkhoff normal form for partial differential equations with tame modulus
• Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
• Perturbations of the defocusing nonlinear Schrödinger equation
• Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation
• The defocusing NLS equation and its normal form
• A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus
• On invariant tori of full dimension for 1D periodic NLS
• Quasi-Periodic Solutions for 1D Schrödinger Equations with Higher Order Nonlinearity

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