Reducible KAM tori for Degasperis-Procesi equation

The Degasperis-Procesi equation (DP) is a spatial one-dimensional model for nonlinear shallow waters phenomena and it is one of the few known Hamiltonian PDEs which is completely integrable, namely it possesses infinitely many constants of motion. Moreover this equation is quasi-linear, namely the nonlinear terms contain derivatives of the same order of the linear part. In this talk I will show a recent result of existence and stability of small amplitude quasi-periodic solutions for Hamiltonian perturbations of the DP equation on the circle. This result is based on a combination of Nash-Moser / KAM schemes and pseudo differential calculus techniques. There are several issues in dealing with this problem:
- the equation is completely resonant, meaning that the linear solutions are all periodic, so the existence of the expected quasi-periodic solutions is due to the nonlinear terms;
- the linear dispersion is weak, in the sense that the linear solutions are close to travelling waves, and this makes difficult to impose the non-resonance Melnikov conditions required by the KAM scheme;
- the resonant structure is quite complicated and we need to exploit the integrability of the unperturbed equation to extract the first nonlinear approximate solutions from which the Nash-Moser scheme bifurcates.
This is a joint work with Roberto Feola and Michela Procesi.