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arXiv 0709.2393math.APmath.DS
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非线性薛定谔方程的KAM

KAM for the Non-Linear Schrödinger Equation

L. H. Eliasson, S. B. Kuksin

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摘要：

本文考虑周期边界条件下的$d$维非线性薛定谔方程：$-i\dot u=-Δu+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $其中$V(x)=\sum \hat V(a)e^{i\sc{a,x}}$是解析函数，$\hat V$是实数，$F$是$\Re u$，$\Im u$和$x$中的实数解析函数. (This方程是“实”NLS方程的一个流行模型，其中取代卷积项$V*u$的是势项$Vu$。）对于$\ep=0$，方程是线性的，并且具有时间准周期解$u$、$$ u(t,x)=\sum_{a\in Å}\hat u(a)e^{i(|a|^2+\hat V(a))t}e^{i\sc{a,x}} \quad (|\hat u(a)|>0), $$，其中$Å$是$\Z^d$的任何有限子集。我们将把$ω_a=|a|^2+\hat V(a)$、$a\inÅ$当作某个定义域$U\subset\R^Å$中的自由参数。这是一个无限自由度的哈密顿系统，退化但有外部参数，我们将描述一个KAM理论，在一般条件下，它将有以下结果：如果$|\ep|$足够小，则存在$U$的大子集$U'$，使得对于所有$ω\in U'$，解$u$持续为时间--准-周期解的所有Lyapounov指数等于零，其线性化方程可约为常系数。

英文摘要：

We consider the $d$-dimensional nonlinear Schrödinger equation under periodic boundary conditions: $-i\dot u=-Δu+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is an analytic function with $\hat V$ real, and $F$ is a real analytic function in $\Re u$, $\Im u$ and $x$. (This equation is a popular model for the `real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\ep=0$ the equation is linear and has time--quasi-periodic solutions $u$, $$ u(t,x)=\sum_{a\in Å}\hat u(a)e^{i(|a|^2+\hat V(a))t}e^{i\sc{a,x}} \quad (|\hat u(a)|>0), $$ where $Å$ is any finite subset of $\Z^d$. We shall treat $ω_a=|a|^2+\hat V(a)$, $a\inÅ$, as free parameters in some domain $U\subset\R^Å$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If $|\ep|$ is sufficiently small, then there is a large subset $U'$ of $U$ such that for all $ω\in U'$ the solution $u$ persists as a time--quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients.

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