  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.
