Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential

Abstract

We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T d , d ≥ 1, and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash-Moser iterative scheme as in [5]. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the "separation properties" of the small divisors assuming weaker non-resonance conditions than in [11].

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

“…With further conditions on the nonlinearity-like reversibility or in the Hamiltonian case-the eigenvalues are purely imaginary, and the torus is linearly stable. The present situation is very different with respect to [18], [13]- [16], [8]- [9] and also [27]- [29], [2], where the lack of stability informations is due to the fact that the linearized equation has variable coefficients, and it is not reduced as in Theorem 1.4 below.…”

“…It is thanks to this "Töplitz-in-time" structure that the linear KdV equation (1.19) is transformed into the dynamical system (1.20). Note that in [27] (and also [16], [8], [9]) the analogous transformations have not this Töplitz-in-time structure and stability informations are not obtained.…”

“…Then, for all λ ∈ G n+1 , we define 9) which is well defined because, if condition (1.8) holds then Π n+1 F (u n ) ∈ H s 00 , and, respectively, if (1.13) holds, then Π n+1 F (u n ) ∈ Y ∩H s (hence in both cases L −1 n Π n+1 F (u n ) exists). Note also that in the reversible case h n+1 ∈ X and so u n+1 ∈ X.…”

Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations

“…Then the operator on the quasi-periodic functions 8) associated to the system (5.5), is transformed by A into…”

“…Naturally one does not need to diagonalize a matrix in order to invert it, indeed in the case of Pde's on tori, where the eigenvalues are multiple, the first results have been proved without any reducibility. See for instance Bourgain in [14,15,17], Berti-Bolle in [6,8], Wang [40]. These papers rely on the so called "multi-scale" analysis based on first Mel'nikov condition and geometric properties of "separation of singular sites".…”