Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions

DOI10.1002/cpa.3012zbMath1031.35137arXivmath-ph/0011036OpenAlexW2069723264MaRDI QIDQ4790289

Horng-Tzer Yau, Tai-Peng Tsai

Publication date: 28 January 2003

Published in: Communications on Pure and Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math-ph/0011036

zbMATH Keywords

decaynonlinear Schrödinger equationresonancebound statesasymptotic profiledispersion-dominated solutionsresonance-dominated solution

Mathematics Subject Classification ID

Scattering theory for PDEs (35P25) Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) NLS equations (nonlinear Schrödinger equations) (35Q55)

Related Items (70)

Coordinates at small energy and refined profiles for the nonlinear Schrödinger equation ⋮ Weakly nonlinear time-adiabatic theory ⋮ Asymptotic Stability of Multi-soliton Solutions for Nonlinear Schrödinger Equations ⋮ Stabilization of solutions to nonlinear Schrödinger equations ⋮ Long-time dynamics of variable coefficient modified Korteweg-de Vries solitary waves ⋮ Stability of standing waves for NLS with perturbed Lamé potential ⋮ Self-similar solutions and collective coordinate methods for nonlinear Schrödinger equations ⋮ STABLE DIRECTIONS FOR EXCITED STATES OF NONLINEAR SCHRÖDINGER EQUATIONS ⋮ Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers ⋮ On Stability of Standing Waves of Nonlinear Dirac Equations ⋮ On stabilization of small solutions in the nonlinear Dirac equation with a trapping potential ⋮ Revisiting asymptotic stability of solitons of nonlinear Schrödinger equations via refined profile method ⋮ Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: neutral modes ⋮ Attractors of Hamilton nonlinear PDEs ⋮ Two-soliton solutions to the three-dimensional gravitational Hartree equation ⋮ On small energy stabilization in the NLKG with a trapping potential ⋮ ON ASYMPTOTIC STABILITY OF GROUND STATES OF NLS ⋮ Nonexistence of small, odd breathers for a class of nonlinear wave equations ⋮ On blowup dynamics in the Keller–Segel model of chemotaxis ⋮ Asymptotic stability of solitons for nonlinear hyperbolic equations ⋮ On traveling waves of the nonlinear Schrödinger equation escaping a potential well ⋮ Quadratic Klein-Gordon equations with a potential in one dimension ⋮ On nonlinear wave equations with parabolic potentials ⋮ A critical center-stable manifold for Schrödinger's equation in three dimensions ⋮ Emission of Cherenkov radiation as a mechanism for Hamiltonian friction ⋮ Asymptotic dynamics of nonlinear Schrödinger equations with many bound states. ⋮ On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation ⋮ Attractors of nonlinear Hamiltonian partial differential equations ⋮ Inelastic interaction of nearly equal solitons for the quartic gKdV equation ⋮ Asymptotic stability of small solitary waves for nonlinear Schrödinger equations with electromagnetic potential in \(\mathbb R^3\) ⋮ The Hamiltonian structure of the nonlinear Schrödinger equation and the asymptotic stability of its ground states ⋮ On the solitary wave dynamics, under slowly varying medium, for nonlinear Schrödinger equations ⋮ On Asymptotic Stability of Solitary Waves in Schrödinger Equation Coupled to Nonlinear Oscillator ⋮ Internal mode-induced growth in \(3d\) nonlinear Klein-Gordon equations ⋮ Kink dynamics in the $\phi ^4$ model: Asymptotic stability for odd perturbations in the energy space ⋮ Small solutions of nonlinear Schrödinger equations near first excited states ⋮ Perturbation expansion and Nth order Fermi golden rule of the nonlinear Schrödinger equations ⋮ Resonance expansions of propagators in the presence of potential barriers. ⋮ Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials ⋮ On asymptotic stability in energy space of ground states of NLS in 1D ⋮ Relaxation of solitons in nonlinear Schrödinger equations with potential ⋮ Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations with time-dependent potential ⋮ Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three ⋮ On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D ⋮ SELECTION OF THE GROUND STATE FOR NONLINEAR SCHRÖDINGER EQUATIONS ⋮ On the asymptotic stability of bound states in 2D cubic Schrödinger equation ⋮ On orbital instability of spectrally stable vortices of the NLS in the plane ⋮ On asymptotic stability of moving ground states of the nonlinear Schrödinger equation ⋮ On asymptotic stability in energy space of ground states of NLS in 2D ⋮ On asymptotic stability of kink for relativistic Ginzburg-Landau equations ⋮ Soliton-Potential Interactions for Nonlinear Schrödinger Equation in $\mathbb{R}^3$ ⋮ Stable directions for small nonlinear Dirac standing waves ⋮ Stability and instability of nonlinear defect states in the coupled mode equations-analytical and numerical study ⋮ On instability of excited states of the nonlinear Schrödinger equation ⋮ ASYMPTOTIC STABILITY OF NONLINEAR SCHRÖDINGER EQUATIONS WITH POTENTIAL ⋮ Long time dynamics of defocusing energy critical \(3+1\) dimensional wave equation with potential in the radial case ⋮ On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations ⋮ Decay of small odd solutions for long range Schrödinger and Hartree equations in one dimension ⋮ On asymptotic stability of nonlinear waves ⋮ A survey on asymptotic stability of ground states of nonlinear Schrödinger equations. II ⋮ Existence and Stability of Schrödinger Solitons on Noncompact Manifolds ⋮ Soliton dynamics for the nonlinear Schrödinger equation with magnetic field ⋮ Stability of small solitary waves for the one-dimensional NLS with an attractive delta potential ⋮ Solitary wave dynamics in an external potential ⋮ Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity ⋮ Numerical studies of stabilized Townes solitons ⋮ Effective dynamics of magnetic vortices ⋮ On selection of standing wave at small energy in the 1D cubic Schrödinger equation with a trapping potential ⋮ Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry ⋮ The ground state of a Gross-Pitaevskii energy with general potential in the Thomas-Fermi limit

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• On the bound states of the nonlinear Schrödinger equation with a linear potential
• Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations
• Spectral properties of Schrödinger operators and time-decay of the wave functions
• Propagation estimates for \(N\)-body Schrödinger operators
• Multichannel nonlinear scattering for nonintegrable equations. II: The case of anisotropic potentials and data
• Local decay of scattering solutions to Schrödinger's equation
• Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations
• Soliton dynamics in a potential
• Multichannel nonlinear scattering for nonintegrable equations
• Modulational Stability of Ground States of Nonlinear Schrödinger Equations
• Lyapunov stability of ground states of nonlinear dispersive evolution equations
• Decay estimates for Schrödinger operators
• Minimal escape velocities
• Stabilization of solutions to nonlinear Schrödinger equations
• The \(W^{k,p}\)-continuity of wave operators for Schrödinger operators

This page was built for publication: Asymptotic dynamics of nonlinear Schrödinger equations: Resonance-dominated and dispersion-dominated solutions