Grébert B., Kappeler T. The Defocusing NLS Equation and Its Normal Form

European Mathematical Society, Zürich, Switzerland, 2014. — 172 p. — (EMS Series of Lectures in Mathematics 18). — ISBN13: 978-3037191316.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, this book treats the defocusing nonlinear Schrödinger (dNLS) equation on the circle with a dynamical systems viewpoint. By developing the normal form theory, it is shown that this equation is an integrable partial differential equation in the strongest possible sense. In particular, all solutions of the dNLS equation on the circle are periodic, quasi-periodic or almost-periodic in time and Hamiltonian perturbations of this equation can be studied near solutions far away from the equilibrium. The book is intended not only for specialists working at the intersection of integrable PDEs and dynamical systems but also for researchers farther away from these fields as well as for graduate students. It is written in a modular fashion; each of its chapters and appendices can be read independently of each other.Zakharov-Shabat operators
Fundamental solution
Estimates
Gradients
Discriminant
Spectra
Dirichlet and Neumann spectrum
Periodic spectrum
Potentials of real type
Poisson brackets
Isospectral sets
Canonical coordinate system
Liouville coordinates
Motivation
Potentials of almost real type
Actions
Psi-Functions
Angles
Birkhoff coordinates
Analyticity
Jacobian
Canonical relations
Diffeomorphism property
Normal form
Appendices
Analytic maps
Hamiltonian formalism
Infinite products
Fourier coefficients
Multiplicities of eigenvalues
Miscellaneous lemmas