Abstract
AbstractWe consider a linear Schrödinger equation with a nonlinear perturbation in ℝ3. Assume that the linear Hamiltonian has exactly two bound states and its eigen‐values satisfy some resonance condition. We prove that if the initial data is sufficiently small and 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: The resonance‐dominated solutions decay as t−1/2 or the dispersion‐dominated solutions decay at least like t−3/2. © 2002 John Wiley & Sons, Inc.
Highlights
Let H0 be the Hamiltonian H0 = −∆ + V − e0 with V a smooth localized potential and e0 < 0 the ground state energy to −∆ + V
For any nonlinear bound state, ψt = Qe−iEt is a solution to the nonlinear Schrodinger equation
Since we aim to show that the error between them decay like t−1/2, we have to track the nonlinear ground states with accuracy at least like t−1/2
Summary
It is nature to consider continuity method by assuming that the approximate nonlinear ground states and various estimates on the wave function are known up to time T. We show that these estimates continue to hold up to time T + δT , with δT small but fixed, provided that all estimates are re-adjusted w.r.t. to the new nonlinear ground state at T + δT. From this outline, it seems that the resonance dominated solutions and the radiation dominated solutions occur on equal footing. Weinstein for explaining to us the beautiful idea in the work [12] and, in particular, to call our attention to the toy model in [12] which contains the basic idea of the resonance decaying in a very illuminating way
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