Reducibility of the dispersive Camassa-Holm equation with unbounded perturbations

2022/11/11 by Xiaoping Wu, Ying Fu, Wu, Xiaoping +3
Mathematics · Physics and Astronomy · #35Q51 #37K55 #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2211.06015

openalex publication_date 2022/11/11 · openalex created_date 2022/11/21 · openalex updated_date 2026/08/01

Abstract

Considered herein is the reducibility of the quasi-periodically time dependent linear dynamical system with a diophantine frequency vector ω∈ O₀ ⊂ 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 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.

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