A KAM theorem for infinite-dimensional discrete systems.
A Vey theorem for nonlinear PDE
Darboux transformation for the nonstationary Schrödinger equation.
Drift and diffusion in phase space
Dynamical systems -- past and present.
Elements of a qualitative theory of Hamiltonian PDEs.
Galerkin averaging method and Poincaré normal form for some quasilinear PDEs
We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincaré normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a differential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true solutions is...
Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is such that for any we find a solution and a time such that . Moreover, the time satisfies the polynomial bound .
KAM techniques in PDE
KAM theory for the hamiltonian derivative wave equation
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.
Lindstedt series, ultraviolet divergences and Moser's theorem
Lyapunov center theorem for some nonlinear PDE's : a simple proof
Minimizers of Dirichlet functionals on the -torus and the weak KAM theory
Nonlinear vibrations of completely resonant wave equations
We present recent existence results of small amplitude periodic and quasi-periodic solutions of completely resonant nonlinear wave equations. Both infinite-dimensional bifurcation phenomena and small divisors difficulties occur. The proofs rely on bifurcation theory, Nash-Moser implicit function theorems, dynamical systems techniques and variational methods.
On nonlinear Schrödinger equations
Propagation of Gevrey regularity over long times for the fully discrete Lie Trotter splitting scheme applied to the linear Schrödinger equation
In this paper, we study the linear Schrödinger equation over the d-dimensional torus, with small values of the perturbing potential. We consider numerical approximations of the associated solutions obtained by a symplectic splitting method (to discretize the time variable) in combination with the Fast Fourier Transform algorithm (to discretize the space variable). In this fully discrete setting, we prove that the regularity of the initial datum is preserved over long times, i.e. times that are...
Quasi-periodic solutions of Hamiltonian PDEs
We overview recent existence results and techniques about KAM theory for PDEs.
Quasi-periodic solutions with Sobolev regularity of NLS on with a multiplicative potential
We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on , 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 then the solutions are . The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators...
Regularity of weak KAM solutions and Mañé’s Conjecture
We provide a crash course in weak KAM theory and review recent results concerning the existence and uniqueness of weak KAM solutions and their link with the so-called Mañé conjecture.