Reducible KAM Tori for the Degasperis-Procesi Equation

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 complicatedsymplectic structure, theweak dispersiveeffects of the linear flow and the presence ofstrong resonant interactionsrequire a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for specialwave packetsolutions 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.