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 ω∈O0⊂Rν. This system is derived from linearizing the dispersive Camassa-Holm equation with quasi-linear perturbations at a small amplitude quasi-periodic function. It is shown that there is a set O∞⊂O0 of asymptotically full Lebesgue measure such that for any ω∈O∞, 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−∂xx)−1∂x, which induces the symplectic structure of the dispersive Camassa-Holm equation.