The Defocusing Nonlinear Schrödinger Equation


Panayotis G. Kevrekidis, Dimitri J. Frantzeskakis and Ricardo Carretero-Gonzalez
SIAM 2015, 440 pages
PRICE (PAPERBACK) £94.95 ISBN 978-1-61197-393-8

This book consists of four chapters covering around 400 pages. These comprise an introduction and three chapters dealing with dark solitons in one dimension, vortices in two dimensions and vortex structures in three dimensions.

The book is concerned with various aspects of Bose-Einstein Condensates (BEC) and the related equation, the Gross-Pitaevskii equation (GPE), which is a type of Nonlinear Schrödinger equation (NLS). The GPE is a mean-field model equation.

The chapters comprise the following contents: Chapter 1 provides a detailed introduction to the major contents of the book. It introduces the GPE equation. Finally the chapter covers an overall picture of the contents of the remainder of the book. Chapter 2 is concerned with the theoretical and experimental aspects of BEC as relevant to the one dimensional case. Here dark solitons are the main topic of discussion. It commences with a general background on them and continues with considering dark solitons as they relate to different types of BEC, from quasi one dimensional to multi-component BECs and finite temperature condensates.

Chapter 3 covers the two dimensional case relating to the BEC. This specifically relates to the study of vortices. This begins by considering the two dimensional version of the GPE. Next the stationary nonlinear solutions of the equation are considered using various standard approaches, including pseudo-arclength. These are used to analyse the infinite branches of nonlinear bound states of the equation, each originating from the corresponding modes of the linear problem. The linear stability of the solutions is also analysed. The principle results for the nonlinear states bifurcating from the linear spectrum of the 2D problem are presented. Later parts of the chapter consider various configurations of vortices. The modification of existence, stability and dynamics of vortices caused by the presence of anisotropic BECs is considered. The double–well potentials covered as part of the analysis of the 1D case are generalised to ‘few-well’ potentials in two dimensions.

Chapter 4 continues the generalisation of the previous chapters by focusing on the three dimensional case relating to BECs. This chapter is limited to providing a brief survey of results and describing the phenomenon generated by three dimensional NLS type settings. There is some analysis in the chapter but it is mostly at the qualitative level due to the fact that most of the work is sufficiently complicated as to require numerical methods. The subsequent sections of the chapter introduce the nomenclature relating to three dimensional structures with single BECs: vortex lines and vortex rings. The focus then turns to multi-component BECs where skyrmions are introduced.

This book is very much aimed at the reader who is interested in the latest research in Bose-Einstein Condensates, combined with the theoretical aspects through applications of the Nonlinear Schrödinger equation in various forms. The focusing aspect of the NLS has been well served in terms of publications; this is the first book on the defocusing aspect of the NLS. It is well written and presented with many diagrams and will be useful to the reader who is interested in this field.

John Bartlett CMath MIMA

Book review published directly onto IMA website (November 2017)

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