Asymptotic Dynamics of Nonlinear Schrodinger Equations: Resonance Dominated and Radiation Dominated Solutions
Asymptotic Dynamics of Nonlinear Schrodinger Equations: Resonance Dominated and Radiation Dominated Solutions
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非线性薛定谔方程的渐近动力学：共振主导解和辐射主导解
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
H. Yau
中科院分区：
文献类型：
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作者：
Tai;H. Yau
本文考虑一个在R^3 $中具有小的非线性扰动的线性Schr“odinger方程.假设线性哈密顿量恰好有两个束缚态，其本征值满足共振条件。我们证明了，如果初始数据在非线性基态附近，则解随着时间趋于无穷大而趋近于某个非线性基态。此外，解非线性薛定谔方程的波函数与其渐近轮廓之间的差可以有两种不同类型的衰减：1.共振主导解衰减为t^{-1/2}$。2.辐射主导的解至少像t^{-3/2}$那样衰减。
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 $t^{-3/2}$.