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The Defocusing NLS Equation and Its Normal Form
EMS Series of Lectures in Mathematics

Benoît Grébert (Université de Nantes, France)
Thomas Kappeler (University of Zürich, Switzerland)

The Defocusing NLS Equation and Its Normal Form

ISBN print 978-3-03719-131-6, ISBN online 978-3-03719-631-1
DOI 10.4171/131
March 2014, 175 pages, softcover, 17 x 24 cm.
32.00 Euro

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 not only intended 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.

Keywords: Defocusing NLS equation, integrable PDEs, normal forms, action and angle variables, Zakharov–Shabat operators

Further Information

Review in Bull. Amer. Math. (2015)

Review in MR 3203027

Review in Zentralblatt MATH