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

2000/11/22 by Tai-Peng Tsai, Horng-Tzer Yau, Tsai, Tai-Peng +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.math-ph/0011036

latex, 71 pages

arxiv created 2001/02/05 · arxiv updated 2009/11/30

Abstract

We consider a linear Schrödinger equation with a small nonlinear perturbation in R³. 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.

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