The Defocusing Nls Equation and Its Normal Form

Book Preface

This book originated from an unpublished, very preliminary manuscript of ours of 2001. Our plan was to use it as a basis for a comprehensive treatment of the defocusing nonlinear Schr¨odinger equation on the circle. Later, J¨urgen P¨oschel joined us in this project and a substantially revised, but in many respects still incomplete version of these notes has been available since 2009. Unfortunately, other commitments prevented J¨urgen from further participating in the project and he decided to withdraw from authorship. We wish to express our appreciation and gratitude for all the contributions he has made and for generously allowing us to use them.

This monograph is concerned with the theory of integrable partial differential equations. It offers a concise case study of the normal form theory of such equations for the defocusing nonlinear Schr¨odinger equation on the circle – one of the most important nonlinear integrable PDEs, both in view of its applications, in particular to nonlinear optics, and of the fact that this equation comes up as an important model equation in more than one space dimension as well.

To be more specific, our starting point is the defocusing nonlinear Schr¨odinger equation on the circle – dNLS for short – considered as an infinite-dimensional integrable system admitting a complete set of independent integrals in involution. We show that dNLS admits a single, global, real-analytic system of coordinates – the cartesian version of action-angle coordinates, also referred to as Birkhoff coordinates, – such that the dNLS Hamiltonian becomes a function of the actions alone. In fact, these coordinates work simultaneously for all Hamiltonians in the dNLS hierarchy.

Similar results were obtained for the Korteweg-de Vries equation (KdV), another important integrable PDE – see [26]. The existence of global Birkhoff coordinates is a special feature of dNLS and KdV. However, for many integrable PDEs, local Birkhoff coordinates may be constructed in parts of phase space satisfying appropriate conditions by developing our approach further. Specifically, in [30] this has been shown for the focusing nonlinear Schr¨odinger equation, which due to the presence of features of hyperbolic dynamics in certain parts of phase space is known not to admit global Birkhoff coordinates.