Asymptotic Dynamics of Nonlinear Schrodinger Equations: Resonance Dominated and Radiation Dominated Solutions

Abstract

We consider a linear Schrödinger equation with a small nonlinear perturbation in $R^3$. Assume that the linear Hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition. We prove that if the initial data is near a nonlinear ground state, then the solution approaches to certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schrödinger equation and its asymptotic profile can have two different types of decay: 1. The resonance dominated solutions decay as $t^{-1/2}$. 2. The radiation dominated solutions decay at least like $t^{-3/2}$.

arXiv commentlatex, 71 pages

References in corpus (1)

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

Cited by in corpus (4)

• On asymptotic stability in energy space of ground states for Nonlinear Schrödinger equations
• Asymptotic stability of small solitons to 1D NLS with potential
• On orbital instability of spectrally stable vortices of the NLS in the plane
• A revision of "On asymptotic stability in energy space of ground states of NLS in 1D"