Review of The Defocusing Nonlinear Schrödinger Equation

Review of The Defocusing Nonlinear Schrödinger Equation
The Defocusing Nonlinear Schrödinger Equation: From Dark Solitons to Vortices and Vortex Rings
by P.G. Kevrekidis, D.J. Frantzeskakis, and R. Carretero-González
429 pp. (2015)
ISBN: : 978-1-611973-93-8
Reviewed by: Mark A. Hoefer
Department of Applied Mathematics
University of Colorado at Boulder
Email: hoefer (at) Colorado.EDU
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.

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