Benoît Grébert

Birkhoff normal form for partial differential equations with tame modulus

We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that we call of Tame Modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions and we use it to study some concrete equations (NLW,NLS) with different boundary conditions. An application to a nonlinear Schrodinger equation on the

KAM for the Quantum Harmonic Oscillator

d

The Defocusing NLS Equation and Its Normal Form

-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular we get lower bounds on the existence time of solutions.

Reconstruction of a potential on the line that is a priori known on the half line

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.

Gaps of One-Dimensional Periodic AKNS Systems

The theme of this monograph is the nonlinear Schrodinger equation. This equation models slowly varying wave envelopes in dispersive media and arises in various physical systems such as water waves, plasma physics , solid state physics and nonlinear optics. More specifically, we consider the defocusing nonlinear Schrodinger (dNLS) equation in one space dimension, iut = - uxx + 2|u|^2u, with periodic boundary conditions. With a viewpoint from infinite dimensional Hamiltonian systems we present a concise and self-contained study of this evolution equation. By developing its normal form theory we show that it is an integrable partial differential equation (PDE) in the strongest possible sense: action--angle coordinates can be constructed which lead to a globally defined coordinate system where the Hamiltonian of the dNLS equation is a function of the actions alone. Actually, this coordinate system simultaneously works for all the Hamiltonians in the dNLS hierarchy. As an immediate consequence it follows that all solutions of the dNLS equation on the circle are periodic, quasi-periodic or almost-periodic in time. Most importantly, such a coordinate system can be used to analyze qualitative properties of the solutions and to study Hamiltonian perturbations of this equation. The book is not only intended for the handful specialists working at the intersection of integrable PDEs and dynamical systems, but also researchers farther away from these fields. In fact, with the aim of reaching out to graduate students as well we have made the book self-containded. In particular we present a detailed study of the spectral theory of (near) self-adjoint Zakharov--Shabat operators on an interval which appear in the Lax pair formulation of the dNLS equation. It is key to the normal form theory of this integrable PDE. Furthermore, the book is written in a modular fashion where each of its chapters as well as its appendices may be read independently of each other.

Resonant dynamics for the quintic nonlinear Schrödinger equation

In this paper the authors study the inverse Schrodinger scattering on the real line. A method is given that allows unique reconstruction of a potential that is a priori known on the half line from the knowledge of the reflection coefficient and the bound state energies. In particular no information on the forming constants is required. The method is based on an appropriate trace formula and on the solution of the nonlinear ordinary differential equation that is obtained when the potential is replaced by its trace formula in the Schrodinger equation. The Deift–Trubowitz approach to inverse scattering is followed. The main new point is the way in which bound states are treated. In addition to its mathematical interest, the case when the potential is a priori known on the half line is particularly interesting in many applications. One can consider for example a potential that has compact support or that it is zero on a half line.

Hamiltonian Interpolation of Splitting Approximations for Nonlinear PDEs

Consider the periodic AKNS operator (see [AKNS] and [Zak-Sha])

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I. Finite-dimensional discretization

\left( {\begin{array}{*{20}{c}} 0 & { - 1} \\ 1 & 0 \\ \end{array} } \right)\frac{d}{{dx}} + \left( {\begin{array}{*{20}{c}} { - q\left( x \right)} & {p\left( x \right)} \\ {p\left( x \right)} & {q\left( x \right)} \\ \end{array} } \right), x \in \mathbb{R}