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Classification of Asymptotic Profiles for Nonlinear Schrödinger Equations with Small Initial Data | | Tai-Peng Tsai
; Horng-Tzer Yau
; | | Date: |
10 May 2002 | | Journal: | Adv. Theor. Math. Phys. 6 (2002) 107-139 | | Subject: | Mathematical Physics; Analysis of PDEs MSC-class: 35Q40, 35Q55 | math-ph math.AP math.MP | | Abstract: | We consider a nonlinear Schrödinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data are localized and small in $H^1$. We prove that exactly three local-in-space behaviors can occur as the time tends to infinity: 1. The solutions vanish; 2. The solutions converge to nonlinear ground states; 3. The solutions converge to nonlinear excited states. We also obtain upper bounds for the relaxation in all three cases. In addition, a matching lower bound for the relaxation to nonlinear ground states was given for a large set of initial data which is believed to be generic. Our proof is based on outgoing estimates of the dispersive waves which measure the relevant time-direction dependent information of the dispersive wave. These estimates, introduced in [16], provides the first general notion to measure the out-going tendency of waves in the setting of nonlinear Schrödinger equations. | | Source: | arXiv, math-ph/0205015 | | Services: | Forum | Review | PDF | Favorites | |
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