Reducible KAM Tori for the Degasperis–Procesi Equation

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. Contents

Traveling Quasi-periodic Water Waves with Constant Vorticity

“…This requires us to keep track along all of the proof of the "momentum conservation property" that we characterize in different ways in Section 3.4. The momentum conservation law has been used in several KAM results for semilinear PDEs since the works [16,17,28,35]; see also [15,20,31] and references therein. The present paper gives a new application in the context of degenerate KAM theory (with additional difficulties arising by the quasi-linear nature of the water waves equations).…”

“…By (2.12), the symplectic form of (2.14) is the standard one, 15) where J −1 is the symplectic operator…”

About linearization of infinite-dimensional Hamiltonian systems

“…It has been early understood that these results might be seen through elaborated versions of "Nash-Moser" theorem see for instance [Bou98,BB15,BCP,CM18]. We finally mention [FGPr,BBHM,BM21,FG] for the case of fully-nonlinear PDEs. See also [BMP21,CY21] and references therein for infinitedimensional tori.…”

Chaotic-Like Transfers of Energy in Hamiltonian PDEs

“…This method is described in Sect. 3 and it is well established in the KAM theory for quasi-linear PDEs (see for instance [2,12]). • Defocusing and Focusing equations: To simplify the exposition, the theorems above only refer to the defocusing Eqs.…”