Philippe Bolle

Periodic Solutions of Nonlinear Wave Equations with General Nonlinearities

AbstractWe present a variational principle for small amplitude periodic solutions, with fixed frequency, of a completely resonant nonlinear wave equation. Existence and multiplicity results follow by min-max variational arguments.

Quasi-periodic solutions with Sobolev regularity of NLS on

We prove the existence of quasi-periodic solutions for Schrodinger equations with a multiplicative potential on T, d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C∞ then the solutions are C∞. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators (“Green functions”) along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and “complexity” estimates.

\mathbb T^d

We consider nonisochronous, nearly integrable, a priori unstable Hamiltonian systems with a (trigonometric polynomial) O(µ)-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time Td = O((1/µ) ln(1/µ)) by a variational method which does not require the existence of “transition chains of tori” provided by KAM theory. We also prove that our estimate of the diffusion time Td is optimal as a consequence of a general stability result derived from classical perturbation theory.  2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved. Resume

with a multiplicative potential

We prove the existence of small amplitude, (2π/ω)-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequencyω belonging to a Cantor-like set of asymptotically full measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser implicit function theorem. In spite of the complete resonance of the equation, we show that we can still reduce the problem to a finitedimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows us to deal also with nonlinearities that are not odd and with finite spatial regularity.

Drift in phase space: a new variational mechanism with optimal diffusion time

Abstract We discuss in the context of nearly integrable Hamiltonian systems a functional analysis approach to the “splitting of separatrices” and to the “shadowing problem”. As an application we apply our method to the problem of Arnold Diffusion for nearly integrable partially isochronous systems improving known results.

Cantor families of periodic solutions for completely resonant nonlinear wave equations

AbstractThis paper deals with perturbed dynamical systems of the form:

Homoclinics and chaotic behaviour for perturbed second order systems

- \ddot u + u = \nabla V\left( u \right) + \varepsilon \nabla _u W\left( {t,u} \right)

Multiplicity of periodic solutions of nonlinear wave equations

where u(t)∈Rn(n⩾1). By means of a variational approach the existence of multibump homoclinics is proved under general assumptions on the Melnikov function. As a particular case, if (W; u) is T-periodic, the existence of approximate and complete Bernoulli shift structures is proved. An application to partial differential equations is also given.