KAM theory for the hamiltonian derivative wave equation

Abstract

We prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations.2000AMS subject classification: 37K55, 35L05.In the last years many progresses have been done concerning KAM theory for nonlinear Hamiltonian PDEs. The first existence results were given by Kuksin [18] and Wayne [29] for semilinear wave (NLW) and Schrödinger equations (NLS) in one space dimension (1d) under Dirichlet boundary conditions, see [24]-[25] and [21] for further developments. The approach of these papers consists in generating iteratively a sequence of symplectic changes of variables which bring the Hamiltonian into a constant coefficients (=reducible) normal form with an elliptic (=linearly stable) invariant torus at the origin. Such a torus is filled by quasi-periodic solutions with zero Lyapunov exponents. This procedure requires to solve, at each step, constant-coefficients linear "homological equations" by imposing the "second order Melnikov" non-resonance conditions. Unfortunately these (infinitely many) conditions are violated already for periodic boundary conditions.In this case, existence of quasi-periodic solutions for semilinear 1d-NLW and NLS equations, was first proved by Bourgain [3] by extending the Newton approach introduced by Craig-Wayne [9] for periodic solutions. Its main advantage is to require only the "first order Melnikov" non-resonance conditions (the minimal assumptions) for solving the homological equations. Actually, developing this perspective, Bourgain was able to prove in [4], [6] also the existence of quasi-periodic solutions for NLW and NLS (with Fourier multipliers) in higher space dimensions, see also the recent extensions in [1], [28]. The main drawback of this approach is that the homological equations are linear PDEs with non-constant coefficients. Translated in the KAM language this implies a non-reducible normal form around the torus and then a lack of informations about the stability of the quasi-periodic solutions.Later on, existence of reducible elliptic tori was proved by Chierchia-You [7] for semilinear 1d-NLW, and, more recently, by Eliasson-Kuksin [12] for NLS (with Fourier multipliers) in any space dimension, see also , Geng-Xu-You [14].An important problem concerns the study of PDEs where the nonlinearity involves derivatives. A comprehension of this situation is of major importance since most of the models coming from Physics are of this kind.In this direction KAM theory has been extended to deal with KdV equations by , , and, for the 1d-derivative NLS (DNLS) and Benjiamin-Ono equations, by Liu-Yuan [22]. The key idea of these results is again to provide only a non-reducible normal form around the torus. However, in this cases, the homological equations with non-constant coefficients are only scalar (not an infinite system as in the Craig-Wayne-Bourgain approach). We remark that the KAM proof is more delicate for DNLS and Benjiamin-Ono, because these equati...

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