Global dynamics above the ground state energy for the cubic NLS equation in 3D
Global dynamics above the ground state energy for the cubic NLS equation in 3D
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DOI：
10.1007/s00526-011-0424-9
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发表时间：
2010-07
影响因子：
2.1
通讯作者：
K. Nakanishi;W. Schlag
中科院分区：
文献类型：
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作者：
K. Nakanishi;W. Schlag
我们将Nakanishi和Schlag(J Difference EQU 250：2299-2333,2011)中关于非线性Klein-Gordon方程的结果推广到三维具有聚焦立方非线性的非线性薛定谔方程，能量的径向数据至多略高于基态。我们证明了初始数据集分裂成九个非空的两两不相交的区域，这些区域具有长时间内解的不同行为：爆破、散射到0或散射到由位相和标度自由度产生的基态族。后一种类型的解形成了一个光滑的中心稳定流形，它包含了基态，并将相空间局部分成两个连通的区域，分别表现出爆破和散射到0。杜伊卡尔茨和鲁登科(Rev Mater Iberoam 26(1)：1-56,2010)所发现的特解，是杜伊卡尔茨和梅勒(Funct anal 18(6)：1787-1840,2009)关于阈值解的开创性工作之后发现的，在这里表现为从基态发出的独特的一维不稳定/稳定流形。类似于Nakanishi和Schlag(J Difference EQU 250：2299-2333,2011)，该证明结合了基态附近的双曲动力学和远离基态的变分结构。证明中的主要技术成分是一个“一次通过”定理，它排除了“几乎同宿轨道”，即那些解从基态的一个小邻域开始，然后离开，最后回到基态的一个小邻域。与Klein-Gordon情况相比，主要的新困难是缺乏有限的传播速度。在维里论证中，我们需要用径向索伯列夫不等式来估计误差。Nakanishi和Schlag(J Different EQU 250：2299-2333,2011)之间的另一个主要区别是需要控制两个调制参数。
We extend the result in Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) on the nonlinear Klein–Gordon equation to the nonlinear Schrödinger equation with the focusing cubic nonlinearity in three dimensions, for radial data of energy at most slightly above that of the ground state. We prove that the initial data set splits into nine nonempty, pairwise disjoint regions which are characterized by the distinct behaviors of the solution for large time: blow-up, scattering to 0, or scattering to the family of ground states generated by the phase and scaling freedom. Solutions of this latter type form a smooth center-stable manifold, which contains the ground states and separates the phase space locally into two connected regions exhibiting blow-up and scattering to 0, respectively. The special solutions found by Duyckaerts and Roudenko (Rev Mater Iberoam 26(1):1–56, 2010), following the seminal work on threshold solutions by Duyckaerts and Merle (Funct Anal 18(6):1787–1840, 2009), appear here as the unique one-dimensional unstable/stable manifolds emanating from the ground states. In analogy with Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011), the proof combines the hyperbolic dynamics near the ground states with the variational structure away from them. The main technical ingredient in the proof is a “one-pass” theorem which precludes “almost homoclinic orbits”, i.e., those solutions starting in, then moving away from, and finally returning to, a small neighborhood of the ground states. The main new difficulty compared with the Klein–Gordon case is the lack of finite propagation speed. We need the radial Sobolev inequality for the error estimate in the virial argument. Another major difference between Nakanishi and Schlag (J Differ Equ 250:2299–2333, 2011) and this paper is the need to control two modulation parameters.