Relaxation of Excited States in Nonlinear Schrödinger Equations

Abstract

We consider a nonlinear Schrödinger equation in ${\mathbb{R}}^3$ with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear {\it excited} state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schrödinger equation approaches to certain nonlinear {\it ground} state as the time tends to infinity.

arXiv commentSubmitted on August 27, 2001

References in corpus (1)

• Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations

Cited by in corpus (7)

• Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
• On asymptotic stability of standing waves of discrete Schrödinger equation in $\mathbb{Z}$
• On instability of excited states of the nonlinear Schrödinger equation
• Long-Time Dynamics of Variable Coefficient mKdV Solitary Waves
• A revision of "On asymptotic stability in energy space of ground states of NLS in 1D"
• Stable Directions for Excited States of Nonlinear Schrödinger Equations
• On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D