# The Defocusing NLS Equation and Its Normal Form

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*Benoît Grébert; Thomas Kappeler*

A publication of the European Mathematical Society

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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in integrable PDEs and dynamical systems.