KAM for Quasi-Linear Hamiltonian Perturbations of the Dispersive Camassa–Holm Equation

The Camassa–Holm (CH) equation, which serves as the dual to the Korteweg–de Vries (KdV) equation, is a completely integrable equation that admits peaked solitons. In this paper, we establish the KAM theory for quasi-linear Hamiltonian perturbations of the dispersive CH equation over the circle. The existence and linear stability of Cantor families of small-amplitude quasi-periodic solutions of this model are proved. Our proof generalizes the arguments for the Degasperis–Procesi equation and makes use of the Birkhoff normal form technique, a Nash–Moser iterative scheme in Sobolev scales and a reduction procedure. Both the Hamiltonian and reversible structures of the equation are fully utilized in our approach. In the reversible case, some new properties of the Birkhoff maps are explored to set up the reversibility of the linearized operator. In addition, a new technique on a presupposed hypothesis of the relation between the perturbed and unperturbed frequencies is proposed to tackle a parameter-independent quasi-linear equation with reversible structure.