Exponential averaging for Hamiltonian evolution equations
Exponential averaging for Hamiltonian evolution equations
复制标题
哈密顿演化方程的指数平均
DOI：
10.1090/s0002-9947-02-03143-4
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发表时间：
2002
影响因子：
1.3
通讯作者：
A. Scheel
中科院分区：
文献类型：
--
作者：
Karsten Matthies;A. Scheel
我们对哈密顿偏微分方程和高频非线性振子之间的非绝热相互作用的大小进行了估计。假定初始条件的空间解析性，我们证明了动力学可以转化为无限维哈密顿系统和非谐振子的非耦合动力学，直到耦合项在振子频率的某一次幂上指数小。这一结果是由Gevrey空间中无限维解析发展方程的抽象平均定理得到的。在Neishtadt对解析常微分方程组的类似结果的基础上，时间估计关键取决于初始条件的空间正则性。结果表明，选择足够平滑的初始数据可以在多大程度上抑制快速强迫和高振荡空间模式之间的强烈共振。一个应用是由一个非线性薛定谔方程组提供的，它耦合到代表小尺度振荡的快速强迫单模。我们给出一个例子，说明我们在这里得到的偏微分方程解的估计必然不同于常微分方程解的估计。
We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrodinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.