d维环面上非线性薛定谔方程的Nekhoroshev型定理
A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the d-dimensional torus.
摘要：
证明了非线性薛定谔方程$$ iu_t=-Δu+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$的Nekhoroshev型定理，其中$$ iu_t=-Δu+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$是一个典型的光滑势，$g$是二元解析的.更精确地说，我们证明了，如果初始数据是解析的宽度$ρ>0$带上的一个限制等于$\eps$，那么，如果$\eps$是足够小的，上述非线性薛定谔方程的解决方案保持解析的宽度$ρ/2$带和有界的$C\eps$带在很长时间的顺序$ \eps^{-α|\ln \eps|^β}$为某些常数$C> 0$，$α>0$和$β<1$。
英文摘要：
We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation $$ iu_t=-Δu+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$ where $V$ is a typical smooth potential and $g$ is analytic in both variables. More precisely we prove that if the initial datum is analytic in a strip of width $ρ>0$ with a bound on this strip equals to $\eps$ then, if $\eps$ is small enough, the solution of the nonlinear Schrödinger equation above remains analytic in a strip of width $ρ/2$ and bounded on this strip by $C\eps$ during very long time of order $ \eps^{-α|\ln \eps|^β}$ for some constants $C> 0$, $α>0$ and $β<1$.
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