Reducibility of the dispersive Camassa-Holm equation with unbounded perturbations

Abstract

Considered herein is the reducibility of the quasi-periodically time dependent linear dynamical system with a diophantine frequency vector $\omega \in {\mathcal{O}}_0\subset {\mathbb{R}}^{\nu }$. 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 $\omega \in {\mathcal{O}}_{\infty }$, 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.

41pages, submitted to Journal of Functional Analysis