Dynamics of Partial Differential Equations

Volume 16 (2019)

Number 1

Reducibility for a class of weakly dispersive linear operators arising from the Degasperis–Procesi equation

Pages: 25 – 94

DOI: https://dx.doi.org/10.4310/DPDE.2019.v16.n1.a2

Authors

R. Feola (Scuola Internazionale Superiore di Studi Avanzati (SISSA),Trieste, Italy)

F. Giuliani (Dipartimento di Matematica, Università degli Studi Roma 3, Roma, Italy)

M. Procesi (Dipartimento di Matematica, Università degli Studi Roma 3, Roma, Italy)

Abstract

We prove reducibility of a class of quasi-periodically forced linear equations of the form\[\partial_t u - \partial_x \circ (1 + a(\omega t, x))u + \mathcal{Q} (\omega t)u = 0 \qquad x \in \mathbb{T} := \mathbb{R}/2 \pi \mathbb{Z},\]where $u = u(t, x), a$ is a small, $C^{\infty}$ function, $\mathcal{Q}$ is a pseudo-differential operator of order $-1$, provided that $\omega \in \mathbb{R}^{\nu}$ satisfies appropriate non-resonance conditions. Such PDEs arise by linearizing the Degasperis–Procesi (DP) equation at a small amplitude quasi-periodic function. Our work provides a first fundamental step in developing a KAM theory for perturbations of the DP equation on the circle. Following [3], our approach is based on two main points: first a reduction in orders based on an Egorov type theorem then a KAM diagonalization scheme. In both steps the key difficulties arise from the asymptotically linear dispersion law. In view of the application to the nonlinear context we prove sharp tame bounds on the diagonalizing change of variables. We remark that the strategy and the techniques proposed are applicable for proving reducibility of more general classes of linear pseudo differential first order operators.

Keywords

Degasperis–Procesi equation, reducibility, Hamiltonian PDEs, pseudo-differential operators, linear first-order operators, quasi-periodic solutions

2010 Mathematics Subject Classification

Primary 35Fxx, 35Lxx, 35Sxx. Secondary 37Kxx.

Received 3 July 2018

Published 5 December 2018