论文标题
带有无限扰动的色散Camassa-Holm方程的降低
Reducibility of the dispersive Camassa-Holm equation with unbounded perturbations
论文作者
论文摘要
本文考虑的是与二磷频率向量向量$ω\ in \ mathcal {o} _0 \ subset \ subset \ mathbb {r}^ν$的差异时间相关的线性动力学系统的降低性。该系统源于在幅度较小的准周期函数下与无界扰动的色散camassa-holm方程线性化。结果表明，有一组$ \ nathcal {o} _ {\ infty} \ subset \ subset \ subset \ nathcal {o} _0 $渐近均匀的lebesgue量度，以至于对于任何$ phoscal in \ nathcal {o} _ {o} _ {\ infty} $，可以固定一个Queffeftient，而不断变化。转型。本文采用的策略由两个步骤组成：（a）基于系统中伪差分运算符的命令的减少，该伪差分运算符将线性化运算符连接到一个具有恒定系数的恒定系数； （b）一种扰动降低性方案，该方案完全对上一步的其余部分进行对角。我们需要解决的可恢复性的主要困难来自操作员$ j =（1- \ partial_ {xx}）^{ - 1} \ partial_ {x} $，它诱导了配件Camassa-Holm方程的符号结构。
Considered herein is the reducibility of the quasi-periodically time dependent linear dynamical system with a diophantine frequency vector $ω\in \mathcal{O}_0 \subset \mathbb{R}^ν$. This system is derived from linearizing the dispersive Camassa-Holm equation with unbounded perturbations at a small amplitude quasi-periodic function. It is shown that there is a set $\mathcal{O}_{\infty} \subset \mathcal{O}_0$ of asymptotically full Lebesgue measure such that for any $ω\in \mathcal{O}_{\infty}$, the system can be reduced to the one with constant coefficients by a quasi-periodic linear transformation. The strategy adopted in this paper consists of two steps: (a) A reduction based on the orders of the pseudo differential operators in the system which conjugates the linearized operator to a one with constant coefficients up to a small remainder; (b) A perturbative reducibility scheme which completely diagonalizes the remainder of the previous step. The main difficulties in the reducibility we need to tackle come from the operator $J=(1-\partial_{xx})^{-1}\partial_{x}$, which induces the symplectic structure of the dispersive Camassa-Holm equation.
