KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation

Abstract

Abstract:We prove the existence of quasi-periodic, small amplitude, solutions for quasi-linear and fully nonlinear forced perturbations of KdV equations. For Hamiltonian or reversible nonlinearities we also obtain the linear stability of the solutions. The proofs are based on a combination of different ideas and techniques: (i) a Nash-Moser iterative scheme in Sobolev scales. (ii) A regularization procedure, which conjugates the linearized operator to a differential operator with constant coefficients plus a bounded remainder. These transformations are obtained by changes of variables induced by diffeomorphisms of the torus and pseudo-differential operators. (iii) A reducibility KAM scheme, which completes the reduction to constant coefficients of the linearized operator, providing a sharp asymptotic expansion of the perturbed eigenvalues.

Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations

“…Important properties of the sets R σ,σ jj are the following. The proofs are quite standard and follow very closely Lemmata 5.2 and 5.3 in [3]. For completeness we give a proof in the Appendix C. Lemma 6.46.…”

“…For s ≥ s 0 H s is a Banach Algebra and H s (T d+1 ) → C(T d+1 ) continuously. As in [3] we consider the frequency vector…”

Quasi-periodic solutions of nonlinear beam equation with prescribed frequencies

“…In Ref. 6 [1][2][3] proved the existence of small amplitude quasi-periodic solutions with prescribed frequency. In this paper, using KAM method, we give the similar answer for the nonlinear beam equation: an advantage of the KAM approach is to provide not only the existence of an invariant torus but also a normal form around it.…”

A note on KAM for gravity-capillary water waves

“…For quasi-linear, also fully nonlinear, perturbations the first KAM results have been recently proved by Baldi-Berti-Montalto in [3], [5], [6] (see also [2], [4]) for Hamiltonian perturbations of Airy, KdV and mKdV equations. These techniques have been applied by Feola-Procesi [18] also to quasi-linear perturbations of 1-d Schrödinger equations and by Montalto [29] to the Kirchhoff equation.…”