一维多势阱Gros-Petaevskii方程的指数时间
Exponential times in the one-dimensional Gross--Petaevskii equation with multiple well potential
摘要：
我们考虑Gross-Petaevskii方程在1空间维与$n$阱捕获势。我们证明，在半经典极限，有限维特征空间相关联的最低n个特征值的线性算子是轻微变形的非线性项到几乎不变的流形M。确切地说，从M或接近M开始的解将保持接近M的指数级时间，与非线性大小的倒数成正比。正如理论上所预期的那样，M上的有效方程是离散非线性Schroedinger方程的扰动。我们推断，当非线性的大小是足够大的，然后隧道之间的井基本上消失：这是几乎所有的解决方案开始接近M的限制，每个井有正常的近似常数在考虑的时间尺度。在双势阱的特殊情况下，我们给出了一个更精确的结果，表明在指数长的时间内，拍运动的持续性或破坏性。证明是基于正则微扰理论;令人惊讶的是，由于系统的规范不变性，不需要非共振条件。
英文摘要：
We consider the Gross-Petaevskii equation in 1 space dimension with a $n$-well trapping potential. We prove, in the semiclassical limit, that the finite dimensional eigenspace associated to the lowest n eigenvalues of the linear operator is slightly deformed by the nonlinear term into an almost invariant manifold M. Precisely, one has that solutions starting on M, or close to it, will remain close to M for times exponentially long with the inverse of the size of the nonlinearity. As heuristically expected the effective equation on M is a perturbation of a discrete nonlinear Schroedinger equation. We deduce that when the size of the nonlinearity is large enough then tunneling among the wells essentially disappears: that is for almost all solutions starting close to M their restriction to each of the wells has norm approximatively constant over the considered time scale. In the particular case of a double well potential we give a more precise result showing persistence or destruction of the beating motions over exponentially long times. The proof is based on canonical perturbation theory; surprisingly enough, due to the Gauge invariance of the system, no non-resonance condition is required.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
