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

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

2000-11-22

arxiv.org/abs/math-ph/0011036v2

Physics

Mathematical Physics

latex, 71 pages

Scientific paper

We consider a linear Schr\"odinger 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 Schr\"odinger 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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