This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose–Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.

https://iopscience.iop.org/article/10.1088/1751-8113/43/21/213001/pdf

Dark solitons in atomic Bose–Einstein condensates: from theory to experiments

This review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose-Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher-dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations.

/pdf/dark-solitons-in-atomic-bose-einstein-condensates-from-5a8fcdp426.pdf

Dark solitons in atomic Bose-Einstein condensates: from theory to experiments

The aim of this review is to introduce the reader to some of the physical notions and the mathematical methods that are relevant to the study of nonlinear waves in Bose–Einstein condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyse some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g. the linear or the nonlinear limit or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.

/pdf/nonlinear-waves-in-bose-einstein-condensates-physical-o6ouu967m4.pdf

Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques

A method is developed for establishing the exact solvability of nonlinear evolution equations in one space dimension which are linear with constant coefficient in the highest‐order derivative. The method, based on the symmetry structure of the equations, is applied to second‐order equations and then to third‐order equations which do not contain a second‐order derivative. In those cases the most general exactly solvable nonlinear equations turn out to be the Burgers equation and a new third‐order evolution equation which contains the Korteweg‐de Vries (KdV) equation and the modified KdV equation as particular cases.

/pdf/a-symmetry-approach-to-exactly-solvable-evolution-equations-ib8nf2zy6z.pdf

A symmetry approach to exactly solvable evolution equations

Determinants, First Minors, and Cofactors.- A Summary of Basic Determinant Theory.- Intermediate Determinant Theory.- Particular Determinants.- Further Determinant Theory.- Applications of Determinants in Mathematical Physics.

http://inis.jinr.ru/sl/m_mathematics/ma_algebra/mal_linear%20algebra/vein%20r.,%20dale%20p.%20determinants%20and%20their%20applications%20in%20mathematical%20physics%20(springer,%201998)(isbn%200387985581)(393s).pdf

Determinants and their applications in mathematical physics

/pdf/b-wave-propagations-in-non-uniform-media-3t3ytwliry.pdf

B. Wave Propagations in Non-Uniform Media

A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex numbers. Operation rules for this transformation is also studied for general number of complex variables. As an application, the max-plus transformation is applied to the discrete Fourier transformation. Stretched coordinates are introduced to obtain the max-plus transformation whose imaginary part coinsides with a phase of the discrete Fourier transformation.

Max-Plus Algebra for Complex Variables and Its Application to Discrete Fourier Transformation

Inverse scattering method is investigated for a general class of evolution equations. A decisive role is played by a transformation operator from the solution of the evolution equation to the eigenfunction for an operator with eigenvalues constant in time. Classes of the evolution equations with the same spectrum problem are presented and a generalization of the Fourier transform method to the nonlinear equation is suggested. The theory is applied to the several evolution equations including the Burgers equation and the vector equation studied by Ablowitz and others. The initial value problem for certain classes of nonlinear evolution equation has been studied by means of the inverse scattering method.D~n In this method, we determine, for a given class of nonlinear equations, the linear operator which depends on the solution of the original equation and, hence, evolves in. time but has invariant spectrum. 2>.B> Thus the time development of the solution of the origi­ nal equation is transferred fully to that of the set of eigenfunctions for linear operator. Since the sets of eigenfunctions determine scattering data in the termino­ logy of quantum mechanics, we are led to the method, where the evolution of the scattering data determines that of the original nonlinear equation. In the present paper we study some classes of the evolution equation and examine the explicit relation between the linear operator for the spectral problem and the evolution equation and also the connection between the solution and the eigenfunctions. As far as the examples known at present are concerned, the evolu­ tion of the eigenfunction is governed by a linear equation. Hence we make a natural proposition that the evolution of the eigenfunctions is the one of the infinitesimal disturbance supposedly taking place in the medium governed by the associated evolution equation. Thus, we assume that the time evolution of the eigenfunction is determined by the Frechet derivative of the evolution equation. In § 2, we consider an evolution equation ut+G[u] =0.,

Spectrum Method for a General Evolution Equation

A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex variables. Operation rules for this transformation is also studied for general number of variables. As an application, the max-plus transformation is applied to the discrete Fourier transformation. Ultradiscretized wave equation and nonlinear Schrodinger equation are solved by the proposed method. The solution has discrete values with inner structures in a discrete space and time.

Max-Plus Algebra for Complex Variables and Its Applications to Discrete Fourier Transformation and Partial Difference Equations

Dans la theorie d'Ablowitz-Kaup-Newell-Segur pour le systeme matriciel N×N, on obtient l'equation d'evolution non lineaire generale associee, dans laquelle l'operateur de recurrence est directement deduit de la condition d'integrabilite. On considere la loi de superposition non lineaire et les symetries associees aux equations d'evolution

/pdf/recursion-operators-for-n-n-matrix-nonlinear-evolution-3d430wlvuh.pdf

Naruyoshi Asano

Papers

B. Wave Propagations in Non-Uniform Media

Max-Plus Algebra for Complex Variables and Its Application to Discrete Fourier Transformation

Spectrum Method for a General Evolution Equation

Max-Plus Algebra for Complex Variables and Its Applications to Discrete Fourier Transformation and Partial Difference Equations

Recursion Operators for N×N Matrix Nonlinear Evolution Equations