00-413 Tai-Peng Tsai and Horng-Tzer Yau

Asymptotic Dynamics of Nonlinear Schr\"odinger Equations: Resonance Dominating and Radiation Dominating Solutions (311K, dvi) Oct 21, 00

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Abstract. 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 dominating solutions decay as $t^{-1/2}$. 2. The radiation dominating solutions decay at least like $t^{-3/2}$.

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