The Defocusing NLS Equation and Its Normal Form
Résumé
The theme of this monograph is the nonlinear Schrödinger 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 Schrödinger (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.