Review of The Defocusing Nonlinear Schrödinger Equation
The nonlinear Schrödinger (NLS) equation
\[
i \psi_t = - \frac{1}{2} \Delta \psi + \sigma |\psi|^2 \psi, \quad (1)
\]
for the complex-valued field \(\psi(\mathbf{x},t)\) encompasses
two very different dispersive, nonlinear wave equations. When \(\sigma
= -1\), Equation (1) is called the
focusing NLS equation
due to the combined effects of nonlinearity and dispersion that
attract or enhance regions of higher intensity or mass density
\(|\psi|^2\). For \(\sigma = +1\), Equation (1) is called the
defocusing NLS equation due to its opposite, repulsive
tendency. The significance of both equations is sourced from the many
areas of physics and mathematics in which they arise due to their
universality: the NLS equation is generically the
asymptotic leading order envelope equation for the propagation of
waves in a weakly nonlinear dispersive medium. Although both
variants of the equation have been studied at length, the defocusing version turns up
36,000 hits on Google whereas the focusing NLS equation has 212,000 hits.
The primary subject of the monograph “The Defocusing Nonlinear
Schrödinger Equation: From Dark Solitons to Vortices and Vortex
Rings” by Kevrekidis, Frantzeskakis, and Carretero-González is
a generalization of the defocusing NLS equation called the
Gross-Pitaevskii equation
\[
i \psi_t = - \frac{1}{2} \Delta \psi + V \psi + |\psi|^2 \psi , \quad (2)
\]
which includes a linear potential \(V(\mathbf{x})\) that can oppose the
repulsive tendency of Equation (1) and confine or trap mass. This
additional effect is a necessity for the study of Bose-Einstein
condensates (BECs) where Equation (2) describes the leading order
evolution of the macroscopic order parameter or wavefunction \(\psi\)
for an ultracold dilute gas of bosons in the lowest energy quantum
state. Due to their macroscopic size, BECs are a veritable playground
for the exploration of nonlinear matter wave phenomena. There has been
an explosion of physical and mathematical research on Equation (2)
since the first observation of a BEC in 1995 and this book surveys
these developments with a view toward connecting them with experiments
where Equation (2) has been a remarkably accurate model. The
authors are key contributors to this field as evidenced by the
approximately 120 out of the book's 810 bibliographic entries that
contain at least one of them (about 35 entries include all three authors).
The organizing theme of this monograph is the effective dimensionality
(one, two, or three spatial dimensions) of the underlying dynamics,
practically achieved in Equation (2) and in experiment by
geometric constraints imposed through the potential \(V.\) After an
introductory chapter, the \(d\)-dimensional case is primarily explored
in Chapter \(d+1\) with an emphasis on corresponding wave solutions to
Equation (2). Chapters 2 (139 pages), 3 (178 pages), and
4 (44 pages) explore dark solitons, vortices, and vortex
lines/rings, respectively. The dark soliton is a localized depression
in density \(|\psi|^2\) that is stationary or moves on a uniform or
slowly varying background. The center of a vortex exhibits zero
density, enclosed by an integral number of \(2\pi\) complex phase
windings. When dynamically stable, both dark solitons and vortices
can exhibit particle-like properties and the authors make copious use
of finite dimensional reductions via multi-scale perturbation theory,
perturbed conservation laws, and effective Lagrangians.
Generally, the dynamics described by Equation (2) involve these
nonlinear wave solutions, fundamental states that have been a
significant focus of BEC research. The primary mathematical tools
utilized are perturbation theory, finite dimensional Hamiltonian
dynamics, stability theory, and numerical methods, with the
progression 1D \(\to\) 3D coinciding with analytical \(\to\) numerical.
Written with applied mathematicians and physicists in mind, this text
embodies the basic tenets of applied mathematics, fluidly
moving between, for example, “negative Krein signature” and
“thermodynamic instability”. The rich description of vortex
dynamics in the presence of a parabolic potential (Chapter 3)
provides a beautiful example. Linearization of Equation (2) is
used as a springboard to both construct nonlinear solutions
bifurcating from the linear limit and to investigate their (spectral)
stability. Through suitable linear combinations of linear
eigenfunctions, the authors show how confined dark solitons and
stripes of differing numbers of vortices on a line originate and are
related. Analogously, ring dark solitons and vortex necklaces are
identified. The linear spectra of nonlinear states yields, in
addition to (in)stability, information about neutrally stable dynamics
such as the displaced vortex precession frequency in the confining
potential \(V\). Moving to the strongly nonlinear regime, the authors
consider the dynamics of a large number of vortices in an effective
finite dimensional particle model for the vortex centers. Here, the
inclusion of the trapping potential \(V\) provides for a novel
generalization of classical fluid dynamics because, in addition to
classical vortex-vortex interactions, each vortex interacts with the
trap resulting in precessional motion. The finite dimensional system
is Hamiltonian and the two-vortex case is integrable, yielding
Lyapunov stability results for guiding center equilibria. The
co-rotating \(n\)-vortex necklace configuration is analyzed and found to
be stable for \(n < 7\) with sufficiently small angular momentum.
Remarkably, these equilibria, for up to \(n = 4\), are observed in experiments
and the finite dimensional model agrees quantitatively. The
connection between co-rotating vortex equilbria and roots of
generating functions and classical polynomials provides a link to and
extension of vortex interactions in classical fluids.
In Chapters 2, 3, and 4, the authors consider
generalizations of Equation (2) to multicomponent, nonlinearly
coupled systems of equations, revealing a diverse zoology of
dark-dark, dark-bright, vortex-vortex, vortex-bright, and dark-bright
ring solitons, some with connections to experiments. Additional
generalizations of Equation (2) include the dimensional
“crossover” regime where a generalized nonlinearity is included and
the addition of linear dissipation, modeling finite temperature
effects in a BEC. While the fully three-dimensional dynamics of
Equation (2) in Chapter 4 do not receive as much attention as
its lower dimensional siblings, the authors utilize modern computing
architectures (GPUs) to numerically explore vortex rings and their
interactions.
This impressive, comprehensive text is a shining example of how
non-trivial mathematical methods can be brought to bear on real
physical problems that are well-described by a mathematical model.
Anyone studying nonlinear waves and dynamics will benefit from soaking
this one in.