The sine-Gordon Model and its Applications: From Pendula and Josephson Junctions to Gravity and High-Energy Physics

The sine-Gordon model is a ubiquitous model of Mathematical Physics with a wide range of applications extending from coupled torsion pendula and Josephson junction arrays to gravitational and high-energy physics models. The purpose of this book is to present a summary of recent developments in this field, incorporating both introductory background material, but also with a strong view towards modern applications, recent experiments, developments regarding the existence, stability, dynamics and asymptotics of nonlinear waves that arise in the model. This book is of particular interest to a wide range of researchers in this field, but serves as an introductory text for young researchers and students interested in the topic. The book consists of well-selected thematic chapters on diverse mathematical and physical aspects of the equation carefully chosen and assigned.

• Language: English
• Publisher: Springer
• Release date: Jul 22, 2014
• ISBN: 9783319067223

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Volume 10

Nonlinear Systems and Complexity

Series EditorAlbert C. J. Luo

For further volumes: http://​www.​springer.​com/​series/​11433

Editors

Jesús Cuevas-Maraver, Panayotis G. Kevrekidis and Floyd Williams

The sine-Gordon Model and its ApplicationsFrom Pendula and Josephson Junctions to Gravity and High-Energy Physics

A319663_1_En_BookFrontmatter_Figa_HTML.png

Editors

Jesús Cuevas-Maraver

Departamento de Físíca Aplicada I, University of Sevilla, Sevilla, Spain

Panayotis G. Kevrekidis

Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA

Floyd Williams

Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA

ISSN 2195-9994e-ISSN 2196-0003

ISBN 978-3-319-06721-6e-ISBN 978-3-319-06722-3

DOI 10.1007/978-3-319-06722-3

Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014944067

© Springer International Publishing Switzerland 2014

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While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Jesús Cuevas-Maraver dedicates this volume to Conchi Caballero for her love and patience during the birth of the book, and to Conchi Cuevas for the joy she gives to his life.

Panayotis G. Kevrekidis dedicates this volume to Maria Kevrekidis, for all her sacrifices that made this book possible, to Despina and Athena Kevrekidis for all the smiles they bring to his life and to George and Despina Kevrekidis for expecting this book, before it was even conceived.

Floyd Williams dedicates this volume to his mother, Mrs. Lee Edna Rollins, who at age 92 remains a source of Godly wisdom to him in matters great and small.

Preface

The sine-Gordon equation is, arguably, one of the most popular nonlinear wave models. This feature is due to a number of appealing traits of this system. One such characteristic is that the sine-Gordon equation arises in a very diverse range of applications. These started as early as the 1860s when it was discovered in the course of the study of surfaces of constant negative curvature through the so-called Gauss–Codazzi conditions for surfaces of curvature − 1 in 3-space. It acquired renewed interest due to the classical study of Frenkel and Kontorova in the context of the discrete analog of the model in the 1930s in the theory of crystal dislocations. Subsequently, the work of A.C. Scott produced a mechanical analog of the system, through the realization of an array of coupled torsion pendula that proved extremely useful both in its visualization and in the experimental observation of its solutions. The realm of relevant applications kept expanding through the emergence of Josephson junction arrays and their fluxons, as well as breathers that were intensely studied in the 1990s and early 2000s. At that time there also surfaced a renewed interest in the discrete form of the model and its remarkable characteristics. Relevant applications of the model have continued to expand with recent proposals involving among others the orbits of a string of stars near the inner Lindblad resonance within a galaxy or the evolution of the electromagnetic field on neuronal microtubules.

Another major source of appeal of this equation consists of its intrinsic mathematical beauty and structure. The complete integrability of the model analyzed in detail in the 1970s, coupled with the ability to produce not only kink but especially (a very uncommon feature for PDE models) exact breather solutions, made the sine-Gordon quite unique. Using Bäcklund transformations one is able to generalize such solutions to multi-kink, multi-breather, and kink-breather type mixtures with remarkable robustness and elastic collision properties, again connected to its integrable underlying structure, Lax pair, infinite conservation laws, etc. Not only were these features identified in the classical realm but also semi-classical quantized versions of the model were proposed, the exact quantum scattering matrix was discovered and intriguing dualities to other well-known models such as the Thirring model were revealed. Such connections have continued to expand even in recent years, e.g., among others with a remarkable set of transformations connecting the sine-Gordon with the short pulse equation (SPE), a recent, intriguing model reduction of the Maxwell equations for the description of few cycle optical pulses. Such connections have been recently utilized to demonstrate the integrability of the SPE, the identification of its solutions, to propose integrable discretizations thereof and so on and so forth. It seems rather extraordinary to us that even 150 years since the initial inception of this deceivingly simple-looking nonlinear variant of the wave equation, the sine-Gordon model still continues to provide mathematical surprises, while in parallel continuing to emerge in novel and ever-expanding physical applications.

In light of all of the above, we decided that it would be a good time to attempt to summarize some of the important recent developments in this field, while also capturing some of the historical perspective of the studies of this PDE and the diverse and broad appeal of its applications. It was with that goal in mind that we pursued the creation of this volume, targeting a number of solicited mini-reviews in different areas of mathematical and physical interest where the sine-Gordon has played a key role. Undoubtedly the resulting chapters have a substantial personal flavor and reflect the particular view and interests of the contributors; nevertheless, we hope and do believe that they give a sense of the excitement, interest and stir that this model equation has brought to different sub-disciplines of mathematics and physics.

The first chapter The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons by Malomed presents an overview of the background of the equation, its motivation, and its principal solutions. It also presents a number of particular mathematical topics, such as perturbations to the model and the development of a perturbation theory for its solutions, as well as discrete and higher dimensional variants of the model.

The second chapter sine-Gordon Equation: From Discrete to Continuum by Chirilus-Brückner et al. focuses more particularly on the discrete form of the equation and the transition from that to the well-known continuum limit thereof. This transition is analyzed from the perspective of the fundamental kink and breather solutions, while at the end some very recent variant has been added in the form of $$\mathcal{P}\mathcal{T}$$ -symmetric forms of the sine-Gordon model.

The third chapter Soliton Collisions by Dmitriev and Kevrekidis presents an overview of the richly studied theme of solitonic collisions in the sine-Gordon and related models. The role of internal modes, of various forms of integrability-breaking perturbations, and of other mechanisms such as the radiationless energy exchange is analyzed in this context.

The discrete model is once again the focus of the fourth chapter Effects of Radiation on sine-Gordon Coherent Structures in the Continuous and Discrete Cases by Cisneros-Ake and Minzoni. In that case, the prototypical problem of traveling of kinks in a discrete sine-Gordon medium and their radiation analysis and potential pinning is explored and also two-dimensional standing and traveling structures are proposed as per recent work of the authors.

The fifth chapter Experimental Results for the sine-Gordon Equation in Arrays of Coupled Torsion Pendula by English gives an overview of experimental developments on the sine-Gordon model as captured by its prototypical realization of coupled torsion pendula. Features such as the modulational instability, the famous Peierls–Nabarro barrier, and the realization (and potential decay due to damping) of solitary waves are presented in different configurations such as lines and rings.

The sixth chapter Soliton Ratchets in sine-Gordon-Like Equations by Quintero focuses on a recent theme, namely the realization of ratchet effects in the sine-Gordon context. The role of symmetries in the ratcheting behavior is elucidated and the asymmetric transport of solitary waves in different driven (and damped) variants of the model and relevant applications is explored.

The seventh chapter The sine-Gordon Equation in Josephson-Junction Arrays by Mazo and Ustinov presents another major experimental playground associated with the sine-Gordon model, namely the realm of Josephson junction arrays. The dynamics of fluxons, quantum solitons, and other such states is examined in different types of experimentally relevant configurations.

Finally, the chapters Some Selected Thoughts Old and New on Soliton-Black Hole Connections in 2d Dilaton Gravity, Dressing with Control: Using Integrability to Generate Desired Solutions to Einstein’s Equations, and A Planar Skyrme-Like Model present a different perspective motivated by the sine-Gordon equation by emphasizing its potential relevance to gravity and high energy physics. In the chapter Some Selected Thoughts Old and New on Soliton-Black Hole Connections in 2d Dilaton Gravity, Williams connects sine-Gordon solitons (kinks, oscillating antikinks, breathers, etc.) and static bright solitons to black hole solutions and black hole vacua in dilaton gravity. In the chapter Dressing with Control: Using Integrability to Generate Desired Solutions to Einstein’s Equations, Beheshti and Tahvildar-Zadeh present a variant of the dressing method which they apply to the sine-Gordon equation while subsequently extending it to Einstein equations and Kerr spacetimes. Then in the chapter A Planar Skyrme-Like Model, Cova focuses on a planar Skyrme-like model and presents its solitons and their interactions. Although the chapters Dressing with Control: Using Integrability to Generate Desired Solutions to Einstein’s Equations and A Planar Skyrme-Like Model involve the study of nonlinear sigma models, these models are referred to as harmonic maps in the chapter Dressing with Control: Using Integrability to Generate Desired Solutions to Einstein’s Equations.

We hope that this broad and diverse (although by no means comprehensive) exposition may benefit both young researchers, including graduate students and post-doctoral fellows, and more seasoned researchers in the field. We are certain that the sine-Gordon equation holds many more surprises in store for the future and would be delighted if this book, intended as a rather partial (in both senses of the word) account of its life so far, allows to springboard further developments along this vein.

Jesús Cuevas-Maraver

Panayotis G. Kevrekidis

Floyd Williams

Sevilla, Spain Amherst, MA Amherst, MA

February, 2014

Acronyms

BT

Bäcklund Transformation

DNLS

Discrete Nonlinear Schrödinger

FK

Frenkel–Kontorova

ILM

Intrinsic Localized Mode

IST

Inverse Scattering transform

JJ

Josephson junction

J-T

Jackiw–Teitelboim

KG

Klein–Gordon

NLS

Nonlinear Schrödinger

PDE

Partial Differential Equation

PN

Peierls–Nabarro

sG

sine-Gordon

VA

Variational Approximation

Contents

The sine-Gordon Model:​ General Background, Physical Motivations, Inverse Scattering, and Solitons 1

Boris A. Malomed

sine-Gordon Equation:​ From Discrete to Continuum 31

M. Chirilus-Bruckner, C. Chong, J. Cuevas-Maraver and P. G. Kevrekidis

Soliton Collisions 59

Sergey V. Dmitriev and Panayotis G. Kevrekidis

Effects of Radiation on sine-Gordon Coherent Structures in the Continuous and Discrete Cases 87

Luis A. Cisneros-Ake and A. A. Minzoni

Experimental Results for the sine-Gordon Equation in Arrays of Coupled Torsion Pendula 111

Lars Q. English

Soliton Ratchets in sine-Gordon-Like Equations 131

Niurka R. Quintero

The sine-Gordon Equation in Josephson-Junction Arrays 155

Juan J. Mazo and Alexey V. Ustinov

Some Selected Thoughts Old and New on Soliton-Black Hole Connections in 2d Dilaton Gravity 177

Floyd L. Williams

Dressing with Control:​ Using Integrability to Generate Desired Solutions to Einstein’s Equations 207

Shabnam Beheshti and Shadi Tahvildar-Zadeh

A Planar Skyrme-Like Model 233

Ramón J. Cova

Index261

© Springer International Publishing Switzerland 2014

Jesús Cuevas-Maraver, Panayotis G. Kevrekidis and Floyd Williams (eds.)The sine-Gordon Model and its ApplicationsNonlinear Systems and Complexity1010.1007/978-3-319-06722-3_1

The sine-Gordon Model: General Background, Physical Motivations, Inverse Scattering, and Solitons

Boris A. Malomed¹  

(1)

Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978, Israel

Boris A. Malomed

Email: [e-mail]

Abstract

This chapter offers an overview of the vast research area developed around the sine-Gordon (sG) equation, including solution methods and various nonlinear modes generated by this equation, viz., topological and dynamical solitons (kinks and breathers), cnoidal waves (chains of kinks), and others. Also included is a survey of physical applications of the sG equation, and an outline of the perturbation theory used for the analysis of physical models based on the sG equation but differing from the ideal integrable form of this equation. Topics presented in the chapter, with some details or in a brief form, are the inverse scattering, transform and the perturbation theory based on it, the Bäcklund transform, energy- and momentum-balance methods for the analysis of the soliton dynamics in perturbed versions of the sG equation, the double sG equation, quantum sG systems, multidimensional sG equations, systems of coupled sG equations, and others.

Keywords

Bäcklund transformBreathersCharge-density wavesFrenkel–Kontorova modelInverse scattering transformJosephson junctionsKinksMagnetic solitonsPerturbation theorySelf-induced transparency

1 Exact and Perturbed sine-Gordon Equations: Fundamental Models of the Contemporary Mathematical and Theoretical Physics

One of commonly known classical equations of mathematical physics is the one-dimensional (1D) Klein–Gordon (KG) equation, i.e., the simplest (and, in that sense, universal) linear model for the propagation of dispersive waves:

$$\displaystyle{ \phi _{\mathit{tt}} -\phi _{\mathit{xx}}+\phi = 0, }$$

(1)

where $$\phi \left (x,t\right )$$ is a real wave field, with t and x standing, as usual, for time and the spatial coordinate. The substitution of $$\phi (x,t) =\phi _{0}\exp \left (\mathit{ikx} - i\omega t\right )$$ into Eq. (1) (the formal use of the complex solution of this real equation simplifies the analysis) yields the dispersion equation between the frequency, ω, and wavenumber, k:

$$\displaystyle{ \omega ^{2} = 1 + k^{2}, }$$

(2)

which is characterized by the presence of the gap in frequency spectrum: $$\omega ^{2}(k) \geq \omega _{\mathrm{gap}}^{2} \equiv 1$$ . In the multidimensional geometry, the spatial derivative in Eq. (1) is replaced by the Laplacian, $$\partial ^{2}/\partial x^{2} + \partial ^{2}/\partial y^{2} + \partial ^{2}/\partial z^{2}$$ .

The 1D sine-Gordon (sG) equation is a nonlinear counterpart of Eq. (1):

$$\displaystyle{ \phi _{\mathit{tt}} -\phi _{\mathit{xx}}+\sin \phi = 0. }$$

(3)

Another known form of the 1D sG equation is written in the light-cone coordinates, $$\xi \equiv \left (x + t\right )/2,\ \tau \equiv \left (x - t\right )/2$$ , instead of the laboratory coordinates, x and t:

$$\displaystyle{ \phi _{\xi \tau } =\sin \phi. }$$

(4)

In the latter form, the sG equation was first discovered by French mathematician Jacques Edmond Émile Bour in 1862, as the Gauss-Codazzi equation describing two-dimensional (2D) surfaces with constant negative curvature (pseudospheres) embedded into the 3D Euclidean space [1]. The integrability of the 1D sG equation in the form of Eq. (4), i.e., practically speaking, a possibility to find vast families of exact analytical solutions, was discovered in the nineteenth century too, in the form of Bäcklund transformations, by Swedish mathematician Albert Victor Bäcklund, in the same context of the differential geometry of pseudospheres [2]. The most remarkable solutions to the 1D sG equation are solitons (i.e., solitary waves, which are the main subject of the present chapter). The Bäcklund transformation generates $$\left (n + 1\right )$$ -soliton solutions of the sG equation from the n-soliton ones, thus making it possible to generate an infinite hierarchy of solutions with increasing complexity, starting from the trivial solution, ϕ ≡ 0, which plays the role of the zero-soliton state.

The second advent of the sG equation had occurred in 1939, in the form of the continuous limit of the Frenkel–Kontorova (FK) model, i.e., a celebrated model of dislocations in solid state [3]. It was derived as a discrete sG equation for lattice wave field ϕ n (t) (n is the discrete coordinate):

$$\displaystyle{ \frac{d^{2}\phi _{n}} {\mathit{dt}^{2}} - C\left (\phi _{n+1} +\phi _{n-1} - 2\phi _{n}\right ) +\sin \phi _{n} = 0, }$$

(5)

where C is a constant of the intersite coupling. Then, a similar model was developed for a chain of adsorbed atoms on a metallic surface [4]. Later, many other physical applications of the FK models have been elaborated [5]. The continuous limit of Eq. (5) implies that discrete coordinate n may be considered as a set of discrete values of a continuous one. Then, the substitution of the truncated Taylor expansion:

$$\displaystyle{ \phi \left (n \pm 1\right ) \approx \phi (n) \pm d\phi /\mathit{dn} + (1/2)d^{2}\phi /\mathit{dn}^{2} }$$

(6)

into Eq. (5) yields Eq. (3) with continuous coordinate $$x \equiv n/\sqrt{C}$$ .

The sG equation had received great popularity in the 1970s as a result of the discovery of its integrability by means of the inverse scattering transform (IST) method [6, 7] and development of the understanding of the significance of this equation as a model of many important physical systems [8]. Around that time, the name sine-Gordon was coined, as a pun based on the pattern of KG. Probably, the first paper in the title of which sine-Gordon equation had appeared was published by J. Rubinstein in 1970 [9] (see also a historical review [10] by A. Scott).

Eventually, it has been understood that the 1D sG equation is one of the most fundamental models of modern mathematical physics. This commonly accepted opinion is based on two basic facts. First, the sG equation is one of few relatively simple nonlinear equations solvable by means of the IST. In fact, together with the Korteweg-de Vries and nonlinear Schrödinger (NLS) equations, it belongs to the small set of most fundamental integrable equations [11]. Second, the sG equation is a universal model for media combining the wave dispersion (the same as in the KG equation) and the nonlinearity which is a periodic function of the field variable, i.e., this variable has the meaning of a phase in the respective physical setting (an overview of physical applications of the sG equation is given below).

Of course, this universality may be violated if the nonlinear term contains additional harmonics of the phase variable. The simplest form of the accordingly perturbed version of the sG equation is the double sG equation, which is usually taken in the following form:

$$\displaystyle{ \phi _{\mathit{tt}} -\phi _{\mathit{xx}}+\sin \phi =\epsilon \sin \left (\phi /2\right ) }$$

(7)

(in this case, $$\sin \left (\phi /2\right )$$ actually represents the fundamental harmonic, while sin ϕ is the second harmonic, but it is assumed that coefficient ϵ in front of the fundamental harmonic is a small parameter). The double sG equation was identified as a separate (nonintegrable) model no later than in 1976 [12].

The dispersion relation for the linearized version of Eq. (7) features the frequency gap like the KG equation, cf. Eq. (2): $$\omega ^{2} =\omega _{ \mathrm{gap}}^{2} + k^{2}$$ , with $$\omega _{\mathrm{gap}}^{2} = 1 -\epsilon /2$$ . Another property which the sG equations share with their linear KG counterpart is the invariance with respect to the Lorentz boost,

$$\displaystyle{ x^{{\prime}} = \left (x -\mathit{ct}\right )/\sqrt{1 - c^{2}},\ t^{{\prime}} = \left (t -\mathit{cx}\right )/\sqrt{1 - c^{2}}, }$$

(8)

i.e., the sG equation and its generalizations are relativistically invariant equations (hence they may be used as simple classical-field-theory models).

A commonly known principle states that a physically relevant equation should be derived from the underlying Lagrangian, and should conserve the corresponding Hamiltonian. Here, for the sake of generality, the Lagrangian and Hamiltonian are written for the double sG equation (7):

$$\displaystyle\begin{array}{rcl} L& =& \int _{-\infty }^{+\infty }\left \{\frac{1} {2}\left (\phi _{x}^{2} -\phi _{ t}^{2}\right ) + \left (1-\cos \phi \right ) - 2\epsilon \left [1 -\cos \left (\phi /2\right )\right ]\right \}\mathit{dx},{}\end{array}$$

(9)

$$\displaystyle\begin{array}{rcl} H& =& \int _{-\infty }^{+\infty }\left \{\frac{1} {2}\left (\phi _{x}^{2} +\phi _{ t}^{2}\right ) + \left (1-\cos \phi \right ) - 2\epsilon \left [1 -\cos \left (\phi /2\right )\right ]\right \}\mathit{dx}.{}\end{array}$$

(10)

In addition to the Hamiltonian (energy), the sG equation, including its perturbed version (7), conserves the momentum, whose form is universal (it does not depend on the particular shape of the nonlinearity in the sG equation, nor even on the fact that the nonlinearity is a periodic function of the field variable, ϕ):

$$\displaystyle{ P = -\int _{-\infty }^{+\infty }\phi _{ x}\phi _{t}\mathit{dx}. }$$

(11)

The nonintegrable equation (7) has exactly two dynamical invariants, H and P. However, the integrable equation (3) has an infinite set of conserved quantities, which is a specific feature of integrable equations. Nevertheless, only the conservation of H and P is essential for physical applications, as higher-order invariants do not admit a clear physical interpretation, unlike the energy and momentum.

2 Mathematical Techniques: The Inverse Scattering Transform, Bäcklund Transform, and Others

The integrability of the sG equation in the form of Eq. (3) is based on the possibility to represent it as the compatibility condition of the system of two linear matrix equations for the two-component complex Jost functions, $$\varPsi \left (x,t\right ) = \left \{\varPsi ^{(1)}\left (x,t\right ),\varPsi ^{(2)}\left (x,t\right )\right \}$$ :

$$\displaystyle\begin{array}{rcl} \varPsi _{x}& \,=\,& \frac{i} {2}\lambda \left (\begin{array}{cc} 1& 0\\ 0 & -1 \end{array} \right )\varPsi \,+\, \frac{i} {4}\left (\phi _{x}\,-\,\phi _{t}\right )\left (\begin{array}{cc} 0&1\\ 1 &0 \end{array} \right )\varPsi \,-\,\frac{i} {8\lambda }\left (\begin{array}{cc} \cos \phi & - i\sin \phi \\ i\sin \phi & -\cos \phi \end{array} \right )\varPsi,{}\end{array}$$

(12)

$$\displaystyle\begin{array}{rcl} \varPsi _{t}& \,=\,& -\frac{i} {2}\lambda \left (\begin{array}{cc} 1& 0\\ 0 & -1 \end{array} \right )\varPsi \,-\,\frac{i} {4}\left (\phi _{x}\,-\,\phi _{t}\right )\left (\begin{array}{cc} 0&1\\ 1 &0 \end{array} \right )\varPsi \,-\,\frac{i} {8\lambda }\left (\begin{array}{cc} \cos \phi & - i\sin \phi \\ i\sin \phi & -\cos \phi \end{array} \right )\varPsi,\quad \quad \,\,{}\end{array}$$

(13)

where λ is a (generally, complex) spectral parameter. The IST technique is based on the analysis of the scattering problem for Eq. (12), where coefficient functions ϕ x , ϕ t and sin ϕ are assumed to be localized, i.e., they all decay fast enough at | x | → ∞. The sG equation with periodic boundary conditions, rather than subject to the localization, is integrable too, on the basis of the Lax-pair representation, but the mathematical technique used in that case is completely different, and actually more complex, being based on methods of algebraic geometry for hyperelliptic Riemann surfaces [13].

The localization of coefficient functions ϕ x , ϕ t and sin ϕ makes it possible to define solutions to Eq. (12) for the Jost functions by their asymptotic form at | x | → ∞, where ϕ x , ϕ t and sin ϕ vanish, while cos ϕ becomes equal to + 1 or − 1, hence coefficients of Eq. (12) are asymptotically constant. Accordingly, solutions to the linear ordinary differential equation with constant coefficients reduce to plane waves. In particular, at x → −∞ they can be defined as

$$\displaystyle{ \varPsi ^{(1)} = 0,\ \varPsi ^{(2)} =\exp \left (-\frac{i} {2}\left (\lambda -\frac{1} {4\lambda }\right )x + \frac{i} {2}\left (\lambda +\frac{1} {4\lambda }\right )t\right ). }$$

(14)

The most general asymptotic form of the same solution at x → +∞ may be represented as a linear combination of two independent plane-wave solutions of the respective linear equation with constant coefficients:

$$\displaystyle\begin{array}{rcl} \varPsi ^{(1)}& =& B(\lambda,t)\exp \left ( \frac{i} {2}\left (\lambda -\frac{1} {4\lambda }\right )x + \frac{i} {2}\left (\lambda +\frac{1} {4\lambda }\right )t\right ),{}\end{array}$$

(15)

$$\displaystyle\begin{array}{rcl} \ \varPsi ^{(2)}& =& A(\lambda,t)\exp \left (-\frac{i} {2}\left (\lambda -\frac{1} {4\lambda }\right )x + \frac{i} {2}\left (\lambda +\frac{1} {4\lambda }\right )t\right ).{}\end{array}$$

(16)

In this way, one can define the scattering data for Eq. (12), i.e., the reflection and transmission coefficients, B(λ) and A(λ), introduced in Eqs. (16) and (15), which connect the asymptotic forms of the same solution at x → ±∞. While these coefficients are originally defined for real values of λ, a fundamental property of the scattering problem based on Eq. (12) is the possibility to analytically extend the scattering data and Jost functions into the upper complex half-plane of the spectral parameter, i.e., to Im(λ) > 0. Obviously, the scattering data are determined by the localized physical fields, which induce mixing between the solutions defined by their plane-wave asymptotic forms at x → ±∞—similar to how a localized potential determines the scattering matrix in the usual one-dimensional linear Schrödinger equation. Thus, the localized physical fields, viz., ϕ t , ϕ x and $$\sin \phi \left (x\right )$$ , are effectively mapped into the scattering data in the upper half-plane of λ, determined by these fields. This mapping, together with the solution of the respective inverse problem. i.e., finding the physical field corresponding to a given set of the scattering data, are the core ingredients of the IST technique (in particular, the latter inverse problem had lent the IST technique its name).

In the general case, the inverse problem is formulated in the form of a rather complex system of linear integral equations for the Jost functions, which are usually called Gelfand-Levitan-Marchenko equations [14]. Explicit exact solutions, both for the Jost functions and the underlying physical field, $$\phi \left (x\right )$$ , are available for the most important particular case, when B(λ) vanishes at real values of λ, while A(λ) is fully determined by its zeros in the complex plane, each zero corresponding to a soliton, see below.

The fundamental asset of the IST is the fact that the temporal evolution of the scattering data, which is determined by Eq. (13) for the evolution of the Jost functions, is actually trivial, contrary to the complex evolution of the physical field in the x-space. As well as the asymptotic forms of the solution to Eq. (12), which are given by simple expressions (14) and (16), (15), the evolution of the scattering data is determined by the consideration of the asymptotic limits of Eq. (13) at x → ±∞, where coefficients of the equation become effectively constant, due to the underlying condition of the localization of the physical fields. Substituting the asymptotic expression (14) for the Jost functions into the accordingly simplified asymptotic form of Eq. (13) at x → −∞ demonstrates that this expression satisfies the equation. On the other hand, substituting the asymptotic expressions (15) and (16) for the same Jost-function solution at x → +∞, where the scattering coefficients are assumed to be functions of time, i.e., A = A(λ; t) and $$B = B\left (\lambda;t\right )$$ , into the asymptotic limit form of Eq. (13) with constant coefficients, which is valid at x → ∞, readily gives rise to the following linear evolution equations:

$$\displaystyle{ \frac{\mathit{dA}(\lambda )} {\mathit{dt}} = 0,\ \frac{\mathit{dB}(\lambda )} {\mathit{dt}} = -i\left (\lambda +\frac{1} {\lambda } \right )B(\lambda ). }$$

(17)

Solutions to Eq. (17) are evident:

$$\displaystyle{ A(\lambda;t) = A\left (\lambda;t = 0\right );\ B(\lambda;t) = B\left (\lambda;t = 0\right )\exp \left (-i\left (\lambda +\frac{1} {\lambda } \right )t\right ). }$$

(18)

These solutions are valid both for real values of λ, as well as in the upper complex half-plane of λ, into which the scattering data may be analytically extended, as stated above.

Equations (17) and (18) demonstrate that, as a matter of fact, the IST resembles the classical Fourier-transform technique, which reduces the relatively complex dynamics of wave fields in the coordinate space, governed by linear dispersive equations with constant coefficients [for instance, Eq. (1)], to the trivial evolution of the Fourier transform in the space of wave vectors, quite similar to that described by solution (18). The difference is that the Fourier transform applies to any linear equation with constant coefficients, while the IST method solves only exceptional nonlinear partial differential equations, such as the sG equation.

The IST applies as well to the sG equation written in terms of the light-cone coordinates, see Eq. (4). In fact, the two versions of the IST technique, for Eqs. (3) and (4), can be obtained from each other by a simple transformation.

As mentioned above, the most important component of the scattering data is a set of eigenvalues of λ at which the transmission coefficient A(λ), defined as per Eq. (16), vanishes. It is easy to demonstrate that the Jost functions corresponding to these eigenvalues exponentially vanish at | x | → ∞. Due to the symmetry imposed by the fact that physical field $$\phi \left (x,t\right )$$ is real, it is also possible to demonstrate that these eigenvalues are located, in the upper half-plane of λ, symmetrically with respect to the imaginary axis. Accordingly, they may be isolated eigenvalues corresponding to purely imaginary values of the spectral parameter, or symmetric pairs:

$$\displaystyle{ \lambda _{0} = i\ \mathrm{Im}(\lambda );\ \lambda _{\pm } = i\ \mathrm{Im}(\lambda ) \pm \mathrm{ Re}(\lambda ), }$$

(19)

with Im(λ) > 0. Note that the eigenvalues remain constant in time, according to Eq. (18). Basic exact solutions of the sG equation and their relation to the eigenvalues of the spectral parameter, as well as to other components of the scattering data, are outlined in the next section.

The quantum version of the sG equation for the operator field can be defined too.