# Reducible KAM tori for Degasperis-Procesi equation

## TL;DR

This paper develops a KAM theory for the Degasperis-Procesi equation, addressing the challenges posed by its complex structure and resonances, and introduces novel techniques that could extend to other integrable PDEs.

## Contribution

It introduces a new KAM approach for quasi-linear Hamiltonian PDEs with complex structures, leveraging integrability and resonance analysis, applicable to similar equations like Camassa-Holm.

## Key findings

- Establishment of reducible KAM tori for the Degasperis-Procesi equation.
- Development of a novel analytical framework handling resonances and weak dispersion.
- Potential applicability of the method to other integrable PDEs such as Camassa-Holm.

## Abstract

We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa-Holm equation.