Existence and stability of quasi-periodic solutions for derivative wave equations

Abstract: In this note we present the new KAM result in [3] which proves the existence of Cantorfamilies of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zeroLyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, thisresult is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems.2000AMSsubject classification: 37K55, 35L05. 1 Introduction In the last years many progresses have been obtained concerning KAM theory for nonlinear PDEs,sincethe pioneeringworksofKuksin[17] andWayne[26]for1-dsemilinearwave(NLW) andSchr¨odinger(NLS) equations under Dirichlet boundary conditions, see [19] and [21] for further developments. Theapproachof these papers consists in generating iteratively a sequence ofsymplectic changes of variableswhich 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 withzero Lyapunov exponents. This procedure requires to solve, at each step, constant-coefficients linear“homological equations” by imposing the “second order Melnikov” non-resonance conditions. Unfor-tunately 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, wasfirst proved by Bourgain [6] by extending the Lyapunov-Schmidt decomposition and the Newtonapproach introduced by Craig-Wayne [11] for periodic solutions. Its main advantage is to requireonly the “first order Melnikov” non-resonance conditions (the minimal assumptions) for solving thehomological equations. It has allowed Bourgain to prove [7], [9] also the existence of quasi-periodicsolutions for NLW and NLS (with Fourier multipliers) in any space dimension, see also the recentextensions in Berti-Bolle[4], [5]. The main drawbackofthis approachis that the homologicalequationsare 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 information about the stability of thequasi-periodic solutions. Later on, existence of reducible elliptic tori was proved by Eliasson-Kuksin[13] for NLS (with Fourier multipliers) in any space dimension, see also Procesi-Xu [22].A challenging frontierconcernsPDEs with unbounded nonlinearities, i.e. containingderivatives. Inthis direction KAM theory has been extended for perturbed KdV equations by Kuksin [18], Kappeler-Po¨schel [15], and, for the 1d-derivative NLS (DNLS) and Benjiamin-Ono equations, by Liu-Yuan[14]. We remark that the KAM proof is more delicate for DNLS and Benjiamin-Ono, because theseequations are less “dispersive” than KdV, i.e. the eigenvalues of the principal part of the differentialoperator grow only quadratically at infinity, and not cubically as for KdV. This difficulty is reflectedin the fact that the quasi-periodic solutions in [18], [15] are analytic, while those of [14] are onlyC