Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors

Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors

We present a complete, exact solution of the problem of the magnetic properties of layered superconductors with an infinite number of superconducting layers in parallel fields H 0. Based on a new exact variational method, we determine the type of all stationary points of both the Gibbs and Helm...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Datum:2004
1. Verfasser: Kuplevakhsky, S.V.
Format: Artikel
Sprache:English
Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2004
Schriftenreihe:Физика низких температур
Schlagworte:
Сверхпроводимость и мезоскопические структуры
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/119840
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors / S.V. Kuplevakhsky // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 856-873. — Бібліогр.: 32 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-119840
record_format dspace
spelling irk-123456789-1198402017-06-11T03:05:13Z Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors Kuplevakhsky, S.V. Сверхпроводимость и мезоскопические структуры We present a complete, exact solution of the problem of the magnetic properties of layered superconductors with an infinite number of superconducting layers in parallel fields H 0. Based on a new exact variational method, we determine the type of all stationary points of both the Gibbs and Helmholtz free-energy functionals. For the Gibbs free-energy functional, they are either points of strict, strong minima or saddle points. All stationary points of the Helmholtz free-energy functional are those of strict, strong minima. The only minimizes of both the functionals are the Meissner (0-soliton) solution and soliton solutions. The latter represent equilibrium Josephson vortices. In contrast, non-soliton configurations (interpreted in some previous publications as «isolated fluxons» and «fluxon lattices») are shown to be saddle points of the Gibbs free-energy functional: They violate the conservation law for the flux and the stationarity condition for the Helmholtz free-energy functional. For stable solutions, we give a topological classification and establish a one-to-one correspondence with Abrikosov vortices in type-II superconductors. In the limit of weak interlayer coupling, exact, closed-form expressions for all stable solutions are derived: They are nothing but the «vacuum state» and topological solitons of the coupled static sine-Gordon equations for the phase differences. The stable solutions cover the whole field range 0 < H < ∞ and their stability regions overlap. Soliton solutions exist for arbitrary small transverse dimensions of the system, provided the field H to be sufficiently high. Aside from their importance for weak superconductivity, the new soliton solutions can find applications in different fields of nonlinear physics and applied mathematics. 2004 Article Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors / S.V. Kuplevakhsky // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 856-873. — Бібліогр.: 32 назв. — англ. 0132-6414 PACS: 74.50.+r, 74.80.Dm, 05.45.Yv http://dspace.nbuv.gov.ua/handle/123456789/119840 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Сверхпроводимость и мезоскопические структуры
Сверхпроводимость и мезоскопические структуры
spellingShingle Сверхпроводимость и мезоскопические структуры
Сверхпроводимость и мезоскопические структуры
Kuplevakhsky, S.V.
Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors
Физика низких температур
description We present a complete, exact solution of the problem of the magnetic properties of layered superconductors with an infinite number of superconducting layers in parallel fields H 0. Based on a new exact variational method, we determine the type of all stationary points of both the Gibbs and Helmholtz free-energy functionals. For the Gibbs free-energy functional, they are either points of strict, strong minima or saddle points. All stationary points of the Helmholtz free-energy functional are those of strict, strong minima. The only minimizes of both the functionals are the Meissner (0-soliton) solution and soliton solutions. The latter represent equilibrium Josephson vortices. In contrast, non-soliton configurations (interpreted in some previous publications as «isolated fluxons» and «fluxon lattices») are shown to be saddle points of the Gibbs free-energy functional: They violate the conservation law for the flux and the stationarity condition for the Helmholtz free-energy functional. For stable solutions, we give a topological classification and establish a one-to-one correspondence with Abrikosov vortices in type-II superconductors. In the limit of weak interlayer coupling, exact, closed-form expressions for all stable solutions are derived: They are nothing but the «vacuum state» and topological solitons of the coupled static sine-Gordon equations for the phase differences. The stable solutions cover the whole field range 0 < H < ∞ and their stability regions overlap. Soliton solutions exist for arbitrary small transverse dimensions of the system, provided the field H to be sufficiently high. Aside from their importance for weak superconductivity, the new soliton solutions can find applications in different fields of nonlinear physics and applied mathematics.
format Article
author Kuplevakhsky, S.V.
author_facet Kuplevakhsky, S.V.
author_sort Kuplevakhsky, S.V.
title Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors
title_short Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors
title_full Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors
title_fullStr Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors
title_full_unstemmed Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors
title_sort topological solitons of the lawrence–doniach model as equilibrium josephson vortices in layered superconductors
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2004
topic_facet Сверхпроводимость и мезоскопические структуры
url http://dspace.nbuv.gov.ua/handle/123456789/119840
citation_txt Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors / S.V. Kuplevakhsky // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 856-873. — Бібліогр.: 32 назв. — англ.
series Физика низких температур
work_keys_str_mv AT kuplevakhskysv topologicalsolitonsofthelawrencedoniachmodelasequilibriumjosephsonvorticesinlayeredsuperconductors
first_indexed 2025-07-08T16:45:37Z
last_indexed 2025-07-08T16:45:37Z
_version_ 1837097959792050176
fulltext Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8, p. 856–873 Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices in layered superconductors Sergey V. Kuplevakhsky B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Science of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: kuplevakhsky@ilt.kharkov.ua Received January 22, 2004 We present a complete, exact solution of the problem of the magnetic properties of layered su- perconductors with an infinite number of superconducting layers in parallel fields H � 0. Based on a new exact variational method, we determine the type of all stationary points of both the Gibbs and Helmholtz free-energy functionals. For the Gibbs free-energy functional, they are either points of strict, strong minima or saddle points. All stationary points of the Helmholtz free-energy functional are those of strict, strong minima. The only minimizes of both the functionals are the Meissner (0-soliton) solution and soliton solutions. The latter represent equilibrium Josephson vortices. In contrast, non-soliton configurations (interpreted in some previous publications as «isolated fluxons» and «fluxon lattices») are shown to be saddle points of the Gibbs free-energy functional: They violate the conservation law for the flux and the stationarity condition for the Helmholtz free-energy functional. For stable solutions, we give a topological classification and es- tablish a one-to-one correspondence with Abrikosov vortices in type-II superconductors. In the limit of weak interlayer coupling, exact, closed-form expressions for all stable solutions are de- rived: They are nothing but the «vacuum state» and topological solitons of the coupled static sine-Gordon equations for the phase differences. The stable solutions cover the whole field range 0 � � �H and their stability regions overlap. Soliton solutions exist for arbitrary small transverse dimensions of the system, provided the field H to be sufficiently high. Aside from their importance for weak superconductivity, the new soliton solutions can find applications in different fields of nonlinear physics and applied mathematics. PACS: 74.50.+r, 74.80.Dm, 05.45.Yv 1. Introduction In Ref. 1, concerned with a microscopic model, and Ref. 2, concerned with the phenomenological Law- rence—Doniach (LD) model [3], we have shown that the problem of the minimization of the Gibbs free-en- ergy functional of layered superconductors with an in- finite number of superconducting (S) layers (N � �) in parallel magnetic fields H � 0 admits an exact solu- tion. Advanced mathematical methods, employed in Refs. 1, 2, allowed us to overcome the complications related to mutual dependence of the phases of the S layers � n . (Unfortunately, these complications were not noticed in previous publications on the LD model [4–9], which led to a loss of minimizers.) The main re- sults of Refs. 1, 2 are worth recalling: The set of minimizers derived in Refs. 1, 2 com- prises the topologically trivial Meissner configuration and true soliton (vortex) configurations. As in the case of the Meissner state and Abrikosov vortices (i.e., the topological solitons [10–13] of the Ginzburg—Lan- dau equations) in continuum type-II superconductors, all these configurations are characterized by the con- served topological index Nv � 0 1 2, , ,... (the vortex number, with Nv � 0 for the Meissner state) and the conserved magnetic flux. For this reason, the topolo- gically nontrivial solution with Nv � 1 has been iden- tified as an elementary Josephson vortex in layered su- perconductors at H � 0. Physically, such a solution © Sergey V. Kuplevakhsky, 2004 can be regarded as a bound state of interlayer flux quanta (one flux quantum per insulating layer). We have termed this solution «a vortex plane» because its field distribution has symmetry typical of plane defects. The present paper complements and completes the investigation of Refs. 1, 2 in two major respects. First, we determine the type of all stationary points of the Gibbs free-energy functional, which allows us to clas- sify all of the solutions available in the literature with regard to their stability. In particular, we show that, except for the Meissner solution, all non-soliton solu- tions (such as, e.g., «vortex lattices» [6,7,14]) corre- spond to saddle points. Second, in the limit of weak interlayer coupling, we derive exact, closed-form ana- lytical expressions for the full set of stable solutions with Nv � 0 1 2, , ,.... These solutions cover the whole field range 0 � � �H , as they should, and include the results of Refs. 1, 2 as particular cases. They illustrate all the features of the Meissner state and vortex struc- ture in weakly coupled superconductors (such as, e.g., an overlap of the stability regions and soliton nature of Josephson vortices) and establish true isomorphism with Abrikosov vortices in type-II superconductors. Moreover, they refute the erroneous belief [15,16] that Josephson vortices «do not exist» in small (along the S layers) structures. Although the issue of stability is crucial for the de- termination of the equilibrium vortex structure in lay- ered superconductors, it has not been addressed in any previous publications*. Mathematically, a classifica- tion of stable solutions amounts to the determination of all points of local minima of the energy functionals. A local minimum of the Gibbs free-energy functional of layered superconductors is determined by the rela- tions [18,19] � � � �� �� � � � f d f dy d dy Hn n n n fn n , , , , ; { , , } A A � � � � 0, (1) � f f df dy d f dy d dy d dyn n n n n n n n� � � � � � � � �� � �� , , , , �A A A� � � � �� � � ; , , , , ;H f df dy d dy Hn n n n� , (2) where �� is the first variation of the Gibbs free-en- ergy functional, induced by small variations �fn , �� n , �A of the modulus of the order parameter (fn), the phase (�n) and the vector potential (A), respectively [1,2]. [For example, numerical non-soliton solutions of Ref. 16 satisfy the stationarity condition (1) but do not meet the condition of the minimum (2).] The value of � on the right-hand side of (2) is associated with thermodynamic (observable) Gibbs free energy. The true equilibrium state corresponds to the mini- mum of the thermodynamic Gibbs free energy (i.e., the absolute minimum of the Gibbs free-energy func- tional) at a given H � 0. The rest of the states, satis- fying (1), (2) at a given H � 0, are thermodynami- cally metastable: As an illustration, we refer to the Meissner state of the semi-infinite (along the layers) layered superconductor near the superheating field Hs [1,2]. To eliminate any questions about the actual equi- librium field configurations in layered superconduc- tors at H � 0, we present an explicit mathematical proof that the Meissner (0-soliton) and vortex-plane (soliton) configurations are unique solutions that sa- tisfy the conditions of the minimum (1), (2). More- over, we show that all the minima are strict and strong [19]. For the sake of diversification, we employ a new method of exact minimization of the Gibbs free-energy functional that, in contrast to Refs. 1, 2, does not involve variation with respect to the phases � n : The new method starts directly from the defini- tion (1), (2) and automatically yields the conserva- tion law for the flux and a full set of soliton boundary conditions. For definitiveness, we restrict ourselves to consideration of the popular LD model: Owing to the relationship [2] between the LD model and the true microscopic model of Ref. 1, all the results hold for the latter model as well. The paper is mostly con- cerned with mathematical aspects of the problem; all major derivations are given in full detail. As regards a comparison with the experiment, the reader is referred to Refs. 1, 2. Section 2 of the paper is devoted to exact mini- mization of the Gibbs free-energy functional. In Sec. 2.1 we specify the geometry of the problem, in- troduce the Gibbs free-energy functional of the LD model, �LD, and discuss some of its properties. Using the conditions of the minimum, in Sec. 2.2 we reduce the problem of the minimization of �LD to that of a simpler functional, �, that possesses the same set of minimizers as �LD. In Sec. 2.3 we prove that the sta- tionary points of � belong to two types: those of the minimum and saddle points. From the conditions of the minimum of � we derive the conservation law for Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 857 * Contrary to what is claimed, e.g., in Ref. 17, this issue cannot be resolved by comparing «characteristic length scales» or values of the energy functionals for distinct field configurations. The main problem is to establish whether the compared configurations locally minimize the energy functionals. the flux, which yields the Meissner and soliton (vor- tex-plane) solutions. The uniqueness of these solu- tions as minimizers of both � and �LD is verified by the consideration of the exact lower bounds of the re- lated Helmholz free-energy functional, �H , that ex- plicitly depend on the conserved topological index Nv . In Sec. 2.4 we present a topological classification of the stable solutions and establish isomorphism be- tween vortex planes in layered superconductors at H � 0 and Abrikosov vortices in type-II supercon- ductors. In Sec. 3, in the limit of weak interlayer coupling, we derive exact, closed-form expressions for all stable solutions of the LD model (the Meissner and the soliton, or vortex, ones). The properties of these solu- tions are thoroughly investigated, all major limiting cases are considered. Isomorphism between vortex planes and ordinary Josephson vortices in the single junction is established. The results obtained are summarized and discussed in Sec. 4. In particular, we explain where unstable non-soliton solutions come from. We also draw a com- parison between our approach to layered superconduc- tors and those of other authors. Mathematical flaws of these latter approaches are explicitly pointed out. In Appendix A we discuss analytical properties and the solution of coupled static sine-Gordon (SG) equa- tions for the phase differences, considered in some pre- vious publications. We prove that the Meissner and vortex-plane solutions, derived in Sec. 3, are the unique stable solutions to the SG equations at H � 0. We also establish a relationship to the exact varia- tional principle of Refs. 1, 2. In Appendix B we verify the fulfillment of the Jacobi—Weierstrass—Hilbert sufficient condition for a strong minimum for the ex- act, closed-form solutions of Sec. 3. In Appendix C we draw a comparison between the soliton solutions of Sec. 3 and the non-soliton («lattice») solutions of Refs. 6, 7, 14, which serves as a good illustration of the general results of the main text. 2. Exact minimization of the LD functional 2.1. Formulation of the problem The geometry of the problem is that of Refs. 1, 2: The layering axis is x, with p being the period; the y axis is directed along the S layers, with � � � �L y L being the region occupied by the system [or �� � � � �y , if L � �]. A static, uniform external field is applied along the z axis: H � �( , , )0 0 0H . The case of external current is not considered, i.e., I � 0. Under the assumption of homogeneity along the z axis, we can write the Gibbs free-energy functional of the LD model as �LD n n n n c zf df dy d dy H pH T W, , , , ; ( ) � � � A � � � � � 2 4 � � � � � � � � � � � ��� �dy f y f y T df y dy L L n n n2 4 2 2 12( ) ( ) ( ) ( ) � n � � � � � � �� �2 2 22( ) ( ) ( , ) ( )T d y dy eA np y f yn y n � � � �� � � r T f y f y f y f y yn n n n n n ( ) [ ( ) ( ) ( ) ( ) cos ( )],2 21 2 2 1 1� � � � � � � � � � 4 2 2 2 1 e T T p dx A x y x A x y y H n p np y x� �( ) ( ) ( , ) ( , ) ( ) � � � � � � � !� 2 , (3) �n n n n x n p np y y y e dxA x y, ( ) ( ) ( ) ( ) ( , )� � � � � � ��1 1 1 2� � � � � �"n x n p np y e dxA x y( ) ( , ) ( ) 2 1 . (4) Here � � �c 1; A � ( , , )A Ax y 0 is the vector potential; Wz is the length of the system in the z direction (Wz # �); f yn ( ) [0 1� �f yn ( ) ] and � n y( ) are, re- spectively, the reduced modulus and the phase of the pair potential $ n y( ) in the nth superconducting layer: $ $n n ny T f y y( ) ( ) ( ) exp ( )� " ; " � �n n n% � �1; H Tc( ) is the thermodynamic critical field; r T( ) is a dimensionless phenomenological parameter of the Jo- sephson interlayer coupling; �( )T and �( )T are the Ginzburg—Landau coherence length and the penetra- tion depth, respectively. The local magnetic field h � ( , , )0 0 h obeys the Maxwell equation h x y A x y x A x y y y x( , ) ( , ) ( , ) � � � � � � . (5) The variables fn , � n , and A are subject to standard requirements [19]: fn , � n are supposed to be smooth in the whole interval � � �L y L, whereas A is piece- wise smooth on the domain of definition, because at x np� only the continuity of A can be required. The summation in Eq. (3) runs over all the S layer indices n. To avoid mathematical complications related to the appearance of infinite sums while retaining the prop- erty of the periodicity of the barrier potential and the absence of outer boundaries in the x direction, it is reasonable to compactify the model [20] by imposing periodic boundary conditions on observable quan- tities, 858 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky f fn N n� � , � �n N n N n ny y� � � ��, ,( ) ( )1 1 , h x Np y h x y( , ) ( , )� � , d y dy eA n N p yn N y � � � � � ( ) [( ) , ]2 � � d y dy eA np yn y � ( ) ( , )2 , (6) and proceed to the thermodynamic limit lim N LD LD/N #� � � �� & in the final expressions. (The existence of this limit will be proved in Sec. 2.3.) Using (6), we can write the total flux through the system as � � � � � � � � � � �� �dy dxh x y dy dxh L L n p np n L L n i p n i ( ) ( ) ( ) ( , ) 1 1 p i N x y�� � � 1 ( , ) � � � � � �dy A n N p y A np y L L y y[ [( ) , ] ( , )] � � � � � � � � �� dx A x L A x L n i p n i p i N x x ( ) ( ) [ ( , ) ( , )] 11 � � �� � � � � � � �1 2 1 1 1 e L Ln i n i n i n i i N [ ( ) ( )], ,� � . (7) What we are going to do now is to find all sets of allowed field configurations { , , }fn n� A that at a given H satisfy the condition of the minimum (2), i.e., �LD n n n nf df dy d dy H, , , , ;� � A � � � � � � � � %�LD n n n nf df dy d dy H, , , , ;� � A % � � �min , , , , ;�LD n n n nf df dy d dy H� � A , (8) where { , , }fn n� A belong to sufficiently small neigh- borhood of { , , }fn n� A . 2.2. A new minimization method Our approach is standard [19] and consists in the determination of an exact lower bound of (3) at a given H and finding the field configuration { , , }fn n� A that makes (3) equal to this lower bound. We begin with the stationarity condition (1) for �LD. Variation with respect to fn immediately yields a set of equations f y f y T d f y dy n n n( ) ( ) ( ) ( ) � � �3 2 2 2 � � � �� � r T f y f y yn n n n ( ) [ ( ) ( ) cos ( ),2 2 1 1� � �� �f y yn n n1 1( ) cos ( )],� � � � � �� "2 2 2( ) ( ) ( , ) ( )T d y dy eA np y f yn y n (9) and boundary conditions ' (df dy Ln ) � 0. Variation with respect to Ax leads to the Maxwell equation � � � %� � � h x y y j y j f y f y yn n n n n n ( , ) ( ) ( ) ( )sin ( ), ,4 41 0 1 1� � � , (11) in the regions ( ) ,n p x np� � �1 and the boundary conditions h x L H( , )) � . (12) In Eq. (11), the quantity j yn n, ( )�1 is the density of the Josephson current between the ( )n � 1 th and the nth layers, and j r T p e T T0 2 216� ( ) / ( ) ( )� � � . By variation with respect to Ay , we obtain the Maxwell equation � � � h x y x ( , ) 0 (13) in the regions ( ) ,n p x np� � �1 and the boundary conditions at the S layers h np y h np y( , ) ( , )� � � �0 0 � � � � � pf y e T d y dy eA np yn n y 2 22 2 ( ) ( ) ( ) ( , ) � � . (14) At this point, it is convenient to partially fix the gauge by the condition* A x y A x y A x yx y( , ) , ( , ) ( , )� %0 , (15) which turns Eqs. (5), (11) and (13) into a system of linear inhomogeneous differential equations for A x y( , ): Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 859 * This can be done by the gauge transformation A A* � � +( , ) ( , ) ( , )x y x y x y, , * � �� � ,n ny y e np y( ) ( ) ( , )2 , where ,( , ) ( , )x y dt A t yx C x � �� . � � � � � 2 0 14 A x y y x j f y f y yn n n ( , ) ( ) ( )sin ( )� " , (16) � � � 2 2 0 A x y x ( , ) , (17) with the boundary conditions A np y A np y( , ) ( , )� � �0 0 , (18) � � � � � � � � A x np y A x np y( , ) ( , )0 0 � � � � � pf y e T d y dy eA np yn n 2 22 2 ( ) ( ) ( ) ( , ) � � , (19) ' (� � ) � A x x L H, . (20) From (19), (20), we get the conditions of the vanish- ing of the intralayer currents at the outer boundaries ' (d dy L eA np Ln� ) � ) �2 0( , ) (21) and the conservation law for the total intralayer cur- rent f y d y dy eA np yn n n2 2 0� � � � � �( ) ( ) ( , ) � . (22) In Refs. 1, 2, relation (22) was employed for the minimization of (3) with respect to the phases � n . The solution of (16), (17) under the conditions (18)–(20) is straightforward and has the form [2] A x y j duf u f u u H xn L y n n( , ) ( ) ( )sin ( ) (� " � � � � � � � � ��4 0 1� np e d y dy n) ( ) � � 1 2 � � " � � � ��r T e T f y duf u f u u f n n L y n n n ( ) ( ) ( ) ( ) [ ( )sin ( ) 4 1 2 2 1 � 1 1( )sin ( )]u un" � , (23) ( )n p x np� � �1 , where f y f yn n( ) ( )� � , (24) " � � �" �n n ny y Z( ) ( ) 2� , Zn � ) )0 1 2, , , ..., (25) and the phase differences " % � �n n n� � 1 obey the solvability conditions d y dy duf u f u u epHn J n L y n n " � " � �� � � ��1 2 1 1 1 2 ( ) ( ) ( )sin ( ) � � " � � � � � � ��- � 2 2 1 2 1 1 1 J n n L y n n f y duf u f u u ( ) ( )[ ( )sin ( ) � " �� �f u un n2 2( )sin ( )] � " � � ��1 2 1 f y duf u f u u n n L y n n ( ) ( )[ ( )sin ( ) � " � � � � �f u un n1 1( )sin ( )] . (26) In Eqs. (26), we have introduced the Josephson penetration depth � �J /ej p� �( )8 0 1 2 and a dimen- sionless parameter - �� p/ [2]. By virtue of (24), (25), equations (26) yield ' ( ' ( ' (d dy L d dy L d dy Ln n" ) � " ) % " )�1 , (27) ' (d dy L epH " ) � 2 . (28) For given fn obeying relations (24), equations (26) constitute a system of nonlinear integrodifferential equations for "n , of first order with respect to differ- entiation. The formulation of the boundary value problem for these equations requires imposition of boundary conditions on "n . Appropriate boundary conditions at y � 0 are provided by the symmetry rela- tions (25), i.e., the boundary conditions are " �n nZ( )0 � , Zn � ) )0 1 2, , , ... (29) The main issue now is to find all solutions to (26), (29) that actually make (3) a minimum. As was shown in Refs. 1, 2, this issue is equivalent to exact minimization of (3) with respect to the phases � n . (See also Appendix A.) Below, we present an alterna- tive, simpler method of exact minimization. 860 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky Instead of minimizing with respect to � n , we intro- duce a new functional � via the relation � �LD n n n n n n nf df dy d dy H f df dy H, , , , ; , , , ;� � �A A � � � � � � � � � � ��0 2 2 4 pH T T Wc z( ) ( )� � � � � � � � � �dy d y dy eA np y f y L L n y n n � ( ) ( , ) ( )2 2 2 , (30) where � �0 2 0 4 % � �LD c zH T pNLW ( ) ( ) � (31) is the LD free energy for H � 0. The functional � will be considered on the same class of functions fn , � n , A as the functional �LD: In particular, these functions are supposed to satisfy conditions (14) at the internal boundaries and natural conditions at the outer bound- aries y L� ) . The advantage of the new functional � f df dy Hn n n, , , ;� A � � � � pH T W dy f y T df y dy c z L L n n 2 2 2 2 4 1 2 1 ( ) [ ( )] ( ) ( ) � � � � � � � � � 2 1 2 2 � � � �� � n n r T f y ( ) [ ( ) � � �� �f y f y f y yn n n n n 2 1 12( ) ( ) ( ) cos ( )],� 4 2 2 2 1 e T T p dx A x y x A x y y H n p np y x� �( ) ( ) ( , ) ( , ) ( )� � � � � � � � � � � � � � � � � 2 (32) is that it has simple properties: The minimization of (32) does not require variation with respect to � n . Moreover, a local minimum of � at a given H, min�, provides a lower bound for �LD. Indeed, the functional � is positive, i.e., � f df dy Hn n n, , , ;� A � � � � 0, (33) and its exact lower bound at H � 0, inf � � 0, is achieved on the field configurations f y y y A A d dyn n x y( ) , ( ) ( ), ,% % % �1 0� , , , (34) where ,( )y is an arbitrary smooth function. [Note that the exact lower bound of (3) at H � 0, inf � �LD � 0, is achieved on the same field configu- rations (34).] Because of (33), the continuous func- tional (32) necessarily has a minimum at any H � 0 in the allowed class of functions, specified above [19]. By virtue of (30) and the definition of the minimum, �LD n n n nf df dy d dy H, , , , ;� � A � � � � � � � � � �� �f df dy Hn n n, , , ;� A 0 � � � � �min , , , ;� �f df dy Hn n n� A 0, (35) where the right-hand side of the second inequality pro- vides the desired lower bound for �LD at a given H. To determine min�, we first find all field configu- rations { , , }fn n� A that satisfy the stationarity condi- tion (1). Variation with respect to fn yields the equa- tions f y f y T d f y dy n n n( ) ( ) ( ) ( ) � � �3 2 2 2 � � � �� � r T f y f y yn n n n ( ) [ ( ) ( ) cos ( ),2 2 1 1� � � �f y yn n n1 1( ) cos ( )],� (36) and boundary conditions (10). Variation with respect to Ax leads to the Maxwell equation (11) in the regions ( ) ,n p x np� � �1 and the boundary con- ditions (12). By variation with respect to Ay , we ob- tain the Maxwell equation (13) in the regions ( ) ,n p x np� � �1 and new boundary conditions at the S layers, h np y h np y( , ) ( , )� � �0 0 . (37) Application of (37) to (14), in turn, yields d y dy eA np yn� ( ) ( , )� �2 0. (38) Combining (14) with (13), we get h x y h y( , ) ( )% , (39) which, upon substitution into (11), results in f y y f y yn n n n n n� � � ��1 1 1 1( )sin ( ) ( )sin ( ), ,� � . (40) Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 861 Relations (40) reflect the continuity of the Jo- sephson currents at x np� and constitute a conse- quence ofU( )1 gauge symmetry: They can be obtained directly, by varying (32) with respect to � n [1,2]. In view of (39), the solution of these relations is straightforward: f y f y y y e dxA x yn n n x p ( ) ( ), ( ) ( ) ( , ),% % �� �� 1 0 2" . (41) Note that relations (39), (41) identically satisfy the periodic boundary conditions (6). According to (41), the phases � n obey the finite difference equation � � �n n n� �� � �1 12 0 with the boundary condition � �n n� � "�1 , whose solution is [21] � ,n y n y y( ) ( ) ( )� " � , (42) where ,( )y is an arbitrary smooth function. Our next course of action follows the steps leading to Eqs. (23)–(29): We fix the gauge by (15) and solve the resulting equations for A x y( , ). As a consequence, in addition to (42), we arrive at coupled equations for f and ", f y f y T d f y dy ( ) ( ) ( ) ( ) � � �3 2 2 2 � � � " �r T y f y( ) [ cos ( )] ( )1 0, ' (df dy L) � 0, (43) d y dy duf u u epH J L y " � " � � � ( ) ( )sin ( ) 1 2 2 2 � , (44) " � � �" � � ) )( ) ( ) , , , , ...y y Z Z2 0 1 2� (45) and explicit expressions A x y ep d y dy x ep d y dy ( , ) ( ) ( ) � " � 1 2 1 2 , , (46) h y ep d y dy ( ) ( ) � "1 2 . (47) Any minimizer { , , }fn n� A of (32), at a given H, necessarily satisfies (15) and (41)–(47). At the same time, this minimizer automatically satisfies (9)–(14) and (16)–(28). Moreover, as can be easily verified by direct substitution, � � �LD n nf df dy H f df dy H, , , ; , , , ;� �A A � � � � � � � � �0 � � � � �min , , , ;� �f df dy Hn n n� A 0. (48) According to (35) and the definition of the minimum (8), this means that �LD nf df dy H, , , ;� A � � � � � � � � �inf , , , , ;�LD n n n nf df dy d dy H� � A � � � �min , , , , ;�LD n n n nf df dy d dy H� � A , (49) i.e., any set { , , }fn n� A that in the gauge (15) satisfies (41)–(47) and minimizes (32) is a minimizer of (3). On the other hand, for any set { , ,f }n n� A that in the gauge (15) does not satisfy (41)–(47) we have �LD n n n nf df dy d dy H, , , , ;� � A � � � � � � � � � �� �f df dy Hn n n, , , ;� A 0 � � � � � �min , , , ;� �f df dy Hn n n� A 0 � � � � � �� �f df dy Hn, , , ;� A 0 � � � � ��LD nf df dy H, , , ;� A � � � �min , , , , ;�LD n n n nf df dy d dy H� � A . (50) A strict inequality in the first line of (50) means that the minimizer { , , }fn n� A makes (3) a strict minimum [19]. [Note that the gauge (15) was employed here for the sake of convenience only: It allowed us to ob- tain an explicit solution for A by simple means.] Summarizing, we have proved the following: A set { , , }fn n� A minimizes the LD functional (3) if and only if it is a minimizer of the functional (32) and, hence, necessarily satisfies the symmetry relations (41), with the resulting local magnetic field obeying (39). In the next subsection, we will show that a full set of the minimizers of (32), and, respectively, of (3), comprises the soliton (vortex-plane) solutions and the Meissner (0-soliton) solution [1,2]. 2.3. The conservation law for the flux and soliton boundary conditions To determine all the minimizers of (32), we first re- write this functional as follows: 862 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky � f df dy d dy H, , , ;" " � � � � � � " " � � � � NpWW H f d dy H Wz H z 2 8 4� � � � , , , (51) where �H f df dy d dy , , ," " � � � � � � � � � � � � NpH T W dy f y T df y dy c z L L2 2 2 2 4 1 2 1 ( ) [ ( )] ( ) ( ) � � 2 � � � � � " � " � � � � � � � � � r T y f y d y dy J( ) [ cos ( )] ( ) ( ) 1 2 2 2 2� � � , (52) and the total flux (7), in view of (15) and (41), is given by . �� �� " � " � % " � �N e L L N dy d y dy L L 2 1 20( ) ( ) ( ) � , (53) with �0 � � / e being the flux quantum. Note that the first term on the right-hand side of (51) is merely the energy of the magnetic field in the absence of Jo- sephson coupling. Note that both � and �H explicitly satisfy Le- gendre’s necessary condition of the strong mini- mum [19]: ' ( � � � � 2 2 2 2 2 0 �( ) [ ] ( ) ( ) H df/dy NpH T W Tc z% � , ' ( � � � � 2 2 2 2 4 0 �( ) [ ] ( ) ( ) ( )H c z J d /dy NpH T W r T T " % � , ' ( ' ( ' ( ' ( � � � � � � 2 2 0 � �( ) ( )H H df/dy d /dy d /dy df/dy" � " % . (54) Therefore, all stationary points of �, �H are either strong minima or saddle points. The stationarity condition for both � and �H re- quires that first variations with respect to f and " va- nish [compare with (1)]. Variation with respect to f yields Eqs. (43), as expected. Consider now the first variation of � and �H with respect to ": � � � � � � f d f dy d dy H, , , ;" " � � � � � " " � � � �� � � � � � �� �H zf d f dy d dy HW , , , 4 , (55) � � � � � �H f d f dy d dy , , ," " � � � � NpH T W r Tc z 2 4 ( ) ( ) � � � " � " � � � � � " � � �dy f y y d y dy y L L J 2 2 2 2 ( )sin ( ) ( ) ( )� � � " � 1 8ep d dy L Wz( ) � ��, (56) where the variation of the flux is � � � � � � �� " � " � % "0 1 2 22 3 4 5 5 � �N e L L N dy d y dy L L 2 1 20[ ( ) ( )] ( ) 5 . (57) The requirement of the vanishing of the volume varia- tion in both (55) and (56) yields d y dy f y y J 2 2 2 2 " � " ( ) ( ) sin ( ) � , (58) which is a mere consequence of (44). However, the requirement of stationarity with respect to surface variation [which is proportional to the variation of the flux (57)] is stronger for (56) than for (55). The surface variation in (55) cancels out owing to the boundary conditions (28). In contrast, conditions (28) do not ensure the vanishing of the surface varia- tion in (56), and the stationarity of �H at H � 0 re- quires �� � 0, (59) or, equivalently, � �� " � � � �N dy d y dy L L 0 1 2 0 � ( ) const , (60) where the inequality sign corresponds to H � 0. Note that higher variations of � and �H are equal to each other: � k � �� � k H (k � 2). Thus, all the min- imizers of�H also minimize�. On the other hand, the functional � has no minimizers other than those that simultaneously minimize �H . Indeed, let { , }f " be a minimizer of � in a class of trial functions that admit arbitrary variations ��. Then { , }f " is necessarily a minimizer of � in a subclass of trial functions that sa- tisfy (59). From the condition for the minimum of � � f f df dy d f dy d dy d dy H� � " � " " � " � � � �� � � � , , , ; � " " � � �� f df dy d dy H, , , ; (61) Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 863 [compare with (2)] on this subclass of trial functions, we have �H f f df dy d f dy d dy d dy � � " � " " � " � � � �� � � � , , , � " " � � � � �� �H zf df dy d dy HW , , , 4� � � � � " � " " � " � � � ��H f f df dy d f dy d dy d dy � � � � , , , � " " � � � ��H f df dy d dy , , , 0, (62) which is the condition of the minimum of �H . For this reason, it is sufficient to find all the minimizers of �H . Physically, conditions (59), (60) ensure the stabil- ity of the flux � against any small perturbations, re- presented by the variations " # " � �", which is a manifestation of the Meissner effect [22]. Condi- tions (59), (60) also imply that � plays the role of a thermodynamic variable in (51), which, in turn, al- lows us to identify �H as the Helmholz free-energy functional. Now we will derive the Meissner (0-soliton) and vortex-plane (soliton) boundary conditions [1,2] from the conservation law for the flux (59), (60). As a starting point, we note that all the extremals of �H that satisfy (58), (28) possess the symmetry proper- ties (45). Since trial functions of this type take on only discrete values at y � 0, " �( )0 �Z, the require- ment of continuity of variations �" imposes the con- straint � ��" � �( )0 0Z . (63) Equations (57), (59), in turn, yield � �" � " �( ) ( )L L . (64) On the other hand, relations (45), applied at y L� , by virtue of (63) yield � �" � � " �( ) ( )L L . (65) Combining (64) with (65), we arrive at the condi- tions �" ) �( )L 0. (66) Using (60), we write: " � " � � " � � � ( ) ( ) ( )L L dy d y dy L L 2 1 2� � � " � � � � � � "� � � �� � � � � � �1 2 1 2� � dy d y dy dy d y dy L L L L ( ) ( ) !� � �const 0, (67) where [ ]x and { }x are, respectively, the integer and fractional parts of x [0 1� �{ }x ]. On the other hand, taking account of (45) and (63), (66), we have " � " � � � " � � � ( ) ( ) ( )L L Z L 2 0 � � const . (68) A comparison of (67) with (68) leads to the identifi- cation " � � % " � � � � � � � � ( ) ( )0 1 2 0 � � Z N dy d y dyv L L , (69) � " � % "� � � �� � � !� � � " � � � � ( ) ( ) , ( ) L dy d y dy L L L � � � 1 2 0. (70) Relations (69), (70) are exactly the Meissner (Nv � 0) and vortex-plane (Nv � 1 2, , ...) boundary conditions employed in Refs. 1, 2. The conserved to- pological index Nv � 0 1 2, , , ... has a clear meaning of the number of solitons (i.e., vortex planes or the Josephson vortices) at H � 0. The conserved flux (60), rewritten via the conserved quantities Nv and " �( )L , takes the form � �� � " � � � �N N L v0 | ( )| � . (71) As follows from (71), the solution with given Nv ap- pears when " � � " � " � �( ) , ( ) , ( )L N L Nv v0 0 2� � , (72) which corresponds to the minimum of the density of the Josephson energy [the third term on the right- hand side of (52)] at the boundaries y L� ) . This so- lution vanishes when " ) � ) " �( ) , ( )L Nn v� �0 , (73) which corresponds to the maximum of the density of the Josephson energy at y L� ) and saddle-point in- stability. By (28), conditions (72), (73) determine, respectively, the exact lower and upper bounds of the stability regions of the Meissner (Nv � 0) and vortex (Nv � 1 2, , ...) configurations in the external field H � 0. We will show now that conditions (69), (70) for the SG equations (58) together with (44) actually specify all the minimizers of (52) [and, correspond- ingly, of (51)]. Indeed, let f y( ) be an arbitrary smooth function that for y L L6 �[ , ] satisfies the condi- 864 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky tions 0 1� �f , ' (( )df/dy L) � 0, and f f y0 % min ( ). Using elementary inequalities a b ab� � 2 (a b, � 0) and 7 7 7 7� � �q q q , we derive a sequence of inequalities for (52) with f f� : �H c zf df dy d dy C NpH T W r T , , , ( ) ( ) " " � � � � � � 2 4� � � " � " � � � � � � � � � � �dy f y d y dy L L J 0 2 2 2 1 2 [ cos ( )] ( )� � � "8 8 8 8 8 8 " � �C NpH T W r T f dy y d y dy c z J L L2 0 4 2 ( ) ( ) sin ( ) ( )� � 8 8 8 8 8 8� � � � � " � � � �C NpH T W r T f N Lc z J v 2 1 2 2 0( ) ( ) cos ( )� � , (74) where C NpH T Wc z% � 2 4 ( ) � � � � � � � � � � � � � � �dy f y T df y dy L L 1 2 1 02 2 2 2 [ ( )] ( ) ( ) � . Inequalities of the type (74) are employed in soliton theory to prove the existence and stability of soliton solutions [10–13,23]. In our case, inequality (74) shows that (52) has an exact lower bound in the class of functions { , }f " parameterized by the conserved quantities Nv and " �( )L : inf , , , [ , ( )]�H vf df dy d dy N L" " � � � � " �9 , 9 [ , ( )]N Lv " � � � � � " � � � � 2 1 2 2NpH T W r T f N Lc z J v ( ) ( ) cos ( )*� � , (75) where f* � 0 is the same for all the sets { , ( )}N Lv " � . In view of the continuity of �H [19], the exact lower bound (75) is achieved on the corresponding so- lution { , }f " to (43), (58), (69), and (70). According to Sec. 2.2., this solution represents the desired mini- mizer of (3). Given that for any minimizer �LD nf df dy H, , , ;� A � � � � N f df dy HLD n& �, , , ;A � � � , where &LD � �, we have thus proved the existence of the thermodynamic limit for N # �. For practical applications, we note the first inte- gral of Eqs. (43), (58) that immediately follows from (52): � �2 2 2 2 2 ( ) ( ) T df dy r T d dy J � � � � " � � � � . �� � � � " �f f r T f C2 4 21 2 1( ) cos , (76) where the constant of integration can be determined from the boundary conditions at y L� � : C e p H f L f LJ� � � � � �2 1 2 2 2 2 2 2 4� ( ) ( ) � � " � �r T L f L( ) [ cos ( )] ( )1 2 . (77) 2.4. A topological classification and isomorphism to Abrikosov vortices in type-II superconductors The above results can be given a very clear interpre- tation within the framework of the theory of topologi- cal defects in continuum media [10–13,24]. Consider the thermodynamic LD free energy, obtained by the substitution of a minimizer { , , }f n� A into (3): � � � LD z zH NpWW H H W ( ) � � � �0 2 8 4� � � � � � � � � � NpH T W dy f y T df y dy c z L L2 2 2 2 4 1 2 1 ( ) [ ( )] ( ) ( ) � � 2 � � � � � " � " � � � � � � � � � r T f y y d y dy J( ) ( )[ cos ( )] ( )2 2 2 1 2 � � � . (78) Owing to the symmetry relations (24), (41), (45) and the boundary conditions (10), (28), the density of �LD H( ) is equal at y L� � and y L� � and thus corresponds to the degenerate equilibrium («vacu- um») state, unperturbed by topological defects (so- litons). Mathematically, the boundary of the interval � � �L y L can be considered as a 0-dimensional sphere: S L L0 � � �{ , }. Given that configurations " and " � 2�Z (Z � ) )0 1 2, , , ...) are physically indistin- guishable, we can fix the values " �( )L as in (70) and regard the functions , ( ( ( � � ( ( ( ( ) � % " � � " � � % " � � � � � � �L L L Z dy d y dyn L L 2 1 2 (79) as continuous maps of the boundary into the additive group of the integers, Z: S0 , # Z. (Z is the group of the degeneracy of the equilibrium state, or the or- der-parameter space.) The fact of the existence of to- Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 865 pologically nontrivial maps of this type, realized by soliton solutions, can be expressed in terms of the zeroth homotopy group [10–13,24] ' (�0 M , where the index «0» stands for the boundary S0 and M is the order-parameter space: �0( )Z Z� . (80) The external field H � 0 breaks the symmetry " # �" [see the third term on the right-hand side of (78)]. Therefore, only the values Z Nv% � 0 1 2, , , ... are allowed, with Nv � 0 being the «vacuum» Meiss- ner state. In this way, we arrive at a natural classifica- tion of the minimizers of (3) with respect to the con- served topological (vortex) number Nv . Note that in the case of Abrikosov vortices in con- tinuum type-II superconductors the boundary is topo- logically equivalent to the circle S1, the order-param- eter space is also the circle: M S% 1 [10–13]. Thus, the pertinent homotopy group is the fundamental group of the circle: �1 1( )S � Z. (81) Since in the presence of an external field H � 0 the topological indices for Abrikosov vortices take on the values Z Nv% � 0 1 2, , , ..., relations (80) and (81) es- tablish an isomorphism between the vortex structure in type-II superconductors and that in layered super- conductors at H � 0, with Nv � 0 for the Meissner state and a single Abrikosov vortex (Nv � 1) standing in a one-to-one correspondence with a single vortex plane (not an «isolated fluxon» as is claimed in previ- ous publications [41]). 3. The exact, closed-form solution for r T( ) �� 1 and isomorphism to Josephson vortices in the single junction Equations (43) and (58) with the soliton boundary conditions (69) and (70) can be solved by perturba- tion methods for arbitrary values of the interlayer cou- pling r T( ). However, of particular interest is the limit of weak coupling, r T( ) �� 1. In the case r T( ) �� 1, the zeroth-order solution to (43) has the form f0 1% . Upon substitution into (58), we obtain the well-known static sine-Gordon equation: d y dy y J 2 2 2 1" � " ( ) sin ( ) � . (82) Under the conditions (69), (70), the exact solution to (82) is " � � � � 0 1 2 2 3 4 5 5( ) ( ) ( ),y N y k K k kv J � � 1 2 2am ,(83) dn L k k k k H HJ s � , 0 1 2 2 3 4 5 5 � �1 2 , N m mv � �2 0 1, , , ...; (84) " � � 0 1 2 2 3 4 5 5( ) ,y N y k kv J � � 2 am , (85) dn L k k k H HJ s� , 0 1 2 2 3 4 5 5 � , N m mv � � �2 1 0 1, , , ... , (86) where am ( )u and dn am( ) ( ) ( )u d/du u� are the Jacobian elliptic functions, and K k( )2 is the elliptic integral of the first kind [25]. The stability ranges for the solution (83)–(86) are determined from (28), (72) and (73). They are given by 0 00� � �H H Nv, ; (87) H H H H N N s N v v v� � � � �1 2 2 1 2, , , ... (88) The upper bounds in (88), (89), HNv ( , , , ...)Nv � 0 1 2 , are determined by the implicit equation L N H H K H H N J v s N s N v v v � � � 0 1 2 2 2 3 4 5 5 5 �( ) , , , , ...1 0 1 2 2 2 , (89) where H eps J� �( )� 1 is the superheating field of the Meissner state in a semi-infinite (0 � � �y ) layered superconductor [1,2], and H H HsL s0 % � is the su- perheating field of the Meissner state for L � �. Upon substitution of (83)–(86) into (78) (with f % 1, " % "), one can verify the lower-bound estimates (74), (75). In Appendix B, we show that the solution (83)–(89) satisfies the Jacobi—Weierstrass—Hilbert sufficient condition for a strong minimum [19] of (51) [and hence of (3)]. Note that equation (82) was first analyzed in the context of a single Josephson junction a long time ago [26,27]. It was discussed in numerous subsequent pub- lications [28]. Unfortunately, the complete, exact, closed-form solution (83)–(89), valid for arbitrary values of L � 0 and H � 0, has not been obtained up until now. This situation manifested itself in the ab- sence of any clear mathematical definition of the Josephson vortex at H � 0 and gave rise to the errone- ous belief [15,16] that Josephson vortices «do not ex- ist» for L J�� � . Equations (83)–(89) provide an explicit form for the complete set of minimizers of the LD model (3) with r T( ) �� 1 and generalize the results of Refs. 1, 2. For Nv � 1 2, , ..., they provide a complete set of so- liton solutions to the coupled static SG equations (A1) and establish a one-to-one correspondence be- tween vortex planes in layered superconductors and 866 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky ordinary Josephson vortices in the single junction. (The only difference lies in the definition of the Josephson length � J .) For these reasons, the proper- ties of (83)–(89) are of crucial importance. Equations (83)–(86) reflect a general soliton fea- ture: Solutions (83), (84) with even Nv cannot be continuously transformed into solutions (85), (86) with odd Nv by changing H and vice versa. Solutions with Nv � 1 are pure solitons (vortex planes) only at H H H N s v � ��1 2 2 , which corresponds to the bound- ary conditions (72). In the rest of the regions (88), we have solitons «dressed» by the Meissner field. (The Meissner and the vortex fields cannot be sepa- rated from each other, because the principle of super- position does not apply to the nonlinear equations (A1). Unfortunately, this important issue was not un- derstood in previous publications [29].) Of special interest is the overlap of the regions (87), (88) for N Nv v� and N Nv v� � 1. Owing to this property, the solutions obtained cover the whole field range 0 � � �H , as they should. The overlap practically vanishes for H HN sv �� . Given that all HNv decrease when W L� 2 increases, the overlap is stronger for largeW and can involve several neighbor- ing states. As explained in the Introduction, the ac- tual equilibrium state is the one that corresponds to the minimum of the thermodynamic Gibbs free energy for given H. A transition from the state with N Nv v� to the state N Nv v� � 1with lower Gibbs free energy is a phase transition of the first-order type [1,2]. In particular, the lower critical field Hc1 is de- termined from the requirement that the Gibbs free en- ergy of the state Nv � 1 be equal to that of the Meiss- ner state (Nv � 0) and satisfy the relation H H H HsL s c sL 2 2 1� � � . In the case � J L�� � �, it is given by H H /c s1 2� � [1,2]. Equations (83)–(89) contain corresponding results of Refs. 1, 2 as limiting cases. For example, by mak- ing the change of variable y y L# � and proceeding to the limit L # � in Eqs. (83), (84) with Nv � 0, we obtain the exact Meissner solution in the semi-infinite interval y 6 ��[ , )0 : " � � � � � ( ) exp [ ] y H y/ H H H J s s 4 2 2 arctan � . (90) By proceeding to the limit L # � in Eqs. (85), (86) with Nv � 1, we arrive at the vortex-plane solution in the infinite interval y 6 �� ��( , ): " �( ) exp [ ]y y/ J4 arctan � . (91) When the screening by Josephson currents is negli- gibly small, i.e., (i) for L J�� � and arbitrary H, or (ii) for H Hs �� and arbitrary L, equations (83)–(89) become " � � �( )y N epHyv� 2 � � � ( ) [sin ( ) cos( )] 1 4 2 2 2 2 2 2 N J v e p H epHy epHy epHW � , (92) where N epHW/v � [ ]� [see (69)]. The overlap of states with different Nv now practically vanishes, and the period of the vortex structure for Nv � 1 is P epH� � / , which refutes the claims [15,16] that Josephson vortices «do not exist» in the limit L J�� � . The substitution of (92) into (78) gives the equilib- rium value of the LD free-energy functional (the ther- modynamic free energy): �LD c zH H T pNLW ( ) ( ) � � 2 4� � � � � � � � � 1 1 8 2 2 r T epHW epHW epHW ep HJ ( ) | sin( )| cos ( ) ( )� � � � � � � � . (93) 4. Discussion We have obtained a complete, exact solution of the problem of the magnetic properties of layered super- conductors with an infinite number of superconduct- ing layers (N � �) in parallel fields H � 0, in the ab- sence of transport currents. Based on a new exact variational method (Sec. 2.2, 2.3 and Appendix A), we have determined the type of all stationary points of the Gibbs free-energy functional (3) and the related Helmholtz free-energy functional [derived from (3) by setting H � 0]: For the Gibbs free-energy func- tional, they are either points of strict, strong minima or saddle points. All stationary points of the Helm- holtz free-energy functional are those of strict, strong minima. By evaluating the surface variation of the Helm- holtz free-energy functional, we have found a comp- lete set of stable, equilibrium field configurations: Namely, the Meissner (0-soliton or «vacuum») solu- tion and soliton (vortex-plane) solutions. These solu- tions conserve the flux and realize exact lower bounds of the Helmholtz free-energy functional in the corre- sponding topological sectors. As shown in Appen- dix A, the absence of soliton solutions of the «fluxon» and «lattice» types at H � 0 is due to the boundary conditions on the derivatives of the phase differences [Eqs. (27), (28)] that require the continuity of the lo- cal field at the outer interfaces. Physically, the fact that a vortex plane locally minimizes the free-energy functionals means that, contrary to a widespread be- Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 867 lief [4], the effective interaction between flux quanta positioned in different insulating layers is attractive. Topological methods of Sec. 2.4 establish the true mathematical analogy between the vortex structure in continuum type-II superconductors and that in lay- ered superconductors at H � 0: It consists in an iso- morphism between a single Abrikosov vortex and a single vortex plane. (Note the role of the conservation law for the flux in the derivation of these results: Such conservation laws in nonlinear field theories always yield soliton solutions that minimize the energy func- tionals [10–13].) In the limit of weak interlayer coupling, r T( ) �� 1, we have derived exact, closed-form analytical expres- sions for all stable solutions [Eqs. (83)–(89)]. Solu- tions (83)–(89) explicitly satisfy the SG equations (A1) with a full set of boundary conditions. They meet the Legendre necessary and Jacobi—Weier- strass—Hilbert sufficient conditions for a strong mini- mum [19] and contain the exact results of Refs. 1, 2 as particular limiting cases. Expressions (83)–(89) pro- vide an adequate description of real physical systems with N �� �[ ]- 1 , - �� 1 (see Appendix A), when boun- dary effects along the layering axis can be neglected. We can now answer the question what kind of con- figurations correspond to saddle points of (3). As shown in Sec. 2.3 and Appendix A, all saddle-point configurations are non-soliton and violate the conser- vation law for the flux. Saddle points of the first type appear if one increases the external field H beyond the stability regions (87), (88). (This type of saddle points exists already in the case of a single Josephson junction.) Saddle points of the second type appear as solutions to the boundary value problem (27), (28), (A18) for (A1), where the Zn violate (A22). Non- soliton solutions of this type, interpreted as «vortex lattices», were considered in some previous publica- tions on the LD model [6,7,14]: As shown in Appen- dix C, they are just perturbations of the soliton solu- tions (83)–(86) with Nv � 0. Note that non-soliton numerical solutions [16,17] for finite (N � �) Jo- sephson-junction stacks, interpreted as «isolated flu- xons», belong to the same type: They are character- ized by the condition 1 2 0 � dy d y dy L L n � � " � � � � � � ( ) for all n and thus constitute perturbations of the Meissner solution (83), (84) with Nv � 0. It is instructive to compare our mathematical ap- proach with previous approaches. Both in Refs. 1, 2 and in the present paper, we start by exact mini- mization of the Gibbs free-energy functional. By de- termining a complete set of the minimizers, we arrive at a natural physical interpretation of all relevant mathematical relations and the identification of equi- librium Josephson vortices as topological solitons (vortex planes). [In the weak-coupling limit, they are just the soliton solutions to the SG equations (A1).] In contrast, previous publications on the LD model started with an a priori assumption that the vortex structure in layered superconductors resembled that in continuum type-II superconductors [4–9]. Unfortu- nately, similarities were erroneously sought in spatial distribution of field configurations. However, unlike the true analogy in terms of homotopy theory (Sec. 2.4), any analogy in the configurational space is precluded by fundamental differences between the Gibbs free-energy functional (3) and that of conti- nuum type-II superconductors. We have already pointed out (Refs. 1, 2 and the In- troduction) the inadequacy of mathematical methods of previous publications. It should be added that the necessity of ensuring the vanishing of the surface vari- ation in the stationarity condition for the Helmholtz free-energy functional was not taken into account [4,7–9]: As a result, the conservation law for the flux and soliton solutions were lost. Since exact solvability and soliton solutions of the SG equations (A1) were not noticed, any mathematical definition of the Jo- sephson vortex could not be given. This situation has led to confusion as to what might be called the Josephson vortex at H � 0 even in the simplest case of the single junction: Hence the erroneous claims [15,16] that Josephson vortices «do not exist» for L J�� � . The problem of the stability of the proposed non- soliton configurations (i.e., whether they correspond to any points of minima of the Gibbs free-energy func- tional) has never been posed in previous theoretical publications. (This fact is not surprising in view of the neglect of the conservation law for the flux and ac- companying uncontrolled mathematical approxima- tions*.) Unfortunately, the issue of stability was dis- regarded in numerical simulations [16] for finite (N � �) Josephson-junction stacks as well: Hence a misunderstanding [17] of the profound physical and 868 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky * For example, as shown by Farid [30], field configurations of the type of Refs. 4, 5 do not constitute any solutions in a true mathematical sense. We add that equations for the phase differences (A1) [let alone the more general equations (9), (10), (26)–(28)] were not derived in Refs. 4, 5. mathematical difference between soliton and non- soliton solutions. In summary, the central result of this paper is that equilibrium Josephson vortices in layered supercon- ductors with N � � are topological solitons of the static SG equations for the phase differences. This re- sult should be viewed in the general context of vortex solutions in nonlinear field theories [10–13]. (For example, Abrikosov vortices in continuum type-II su- perconductors are topological solitons of the GL equa- tions.) Mathematically, the exact, closed-form expres- sions (83)–(89) represent a new class of soliton solutions. Aside from their importance for weak super- conductivity, they can find applications in different fields of nonlinear physics and applied mathematics where the SG equations are involved [11]. Appendix A The solution of coupled static SG equations In the limit of weak coupling, considered in Sec. 3, the zeroth-order [with respect to r T( ) �� 1] solution to (9), (10) has the form fn % 1. Upon substitution into Eqs. (26) and subsequent differentiation with respect to y, the latter equations are reduced to coupled static SG equations � - J n m m d y dy G n m y n N2 2 2 2 11 1 " � " ���( ) ( , )sin ( ), ,..., , (A1) where G n m�1( , ) is a Jacobian matrix [31] with elements G n n� � �1 22( , ) - (n N� 1, ..., ), G n n� �1 1( , ) � � � ��G n n1 1 1( , ) (n N� �1 1, ..., ), and G n m� �1 0( , ) for 7 7n m� � 1. Owing to periodic boundary conditions " � "�n N ny y( ) ( ) (A2) [see (6)], the matrix G n m�1( , ) is cyclic. Equations (A1) are subject to conditions (27), (28), and their solutions obey the symmetry relations (25). In the limit N # �, equations (A1) were derived by a different method in a number of publications [6,7]. Unfortunately, their analytical properties have not been studied. The main property can be formu- lated as the following proposition: Proposition. Consider Eqs. (A1) on the whole axis �� � � � �y . The initial value problem for Eqs. (A1) with arbitrary initial conditions " �n ny( )0 : , d dy yn n " �( )0 ; (| |y0 � �) has a unique solution in the whole interval �� � � � �y . This solution has continuous deriva- tives with respect to y of arbitrary order and depends continuously on the initial data. To prove the Proposition, we note that Eqs. (A1) satisfy the conditions of Picard’s theorem on the exis- tence and uniqueness of a global solution [32]. This property is rather unusual for nonlinear differential equations: Global character of the solution and its in- finite differentiability are ensured by the fact that the "n enter the right-hand side of Eqs. (A1) only as argu- ments of the sine. Equations (A1) can be rewritten in an equivalent form: sin ( ) ( , ) ( ) , ,...," � " ��n J m my G n m d y dy n N- �2 2 2 2 1 . (A3) The matrix G n m( , ), being the inverse of G n m�1( , ), has the form 7 7 G n m / / n m ( , ) ,� � � � � � �9 - - 9 - - - 2 1 4 1 2 1 4 2 2 2 , (A4) and obeys the summation rule G n m m � �( , ) 1 2- . (A5) The matrix G n m( , ) is positive definite, because all its eigenvalues ek are positive: ek k J � � - � 2 2 2 , � -� - � k J k/N � � �2 2 22 cos ( ) , k N � ) � � � 0 1 2 , , ..., . (A6) The quantities � k in (A6) are the characteristic length scales of Eqs. (A1). [Note that (A1) are char- acterized by a distribution of length scales, not just two length scales, as is claimed in some previous pub- lications.] The distribution of the length scales be- comes quasicontinuous under the condition N �� �[ ]- 1 , (A7) which can be regarded as a criterion of the applicabil- ity of the LD model to layered superconductors. The fact that equilibrium solutions (83)–(89) cor- respond to the largest length scale � � �max % %0 J is by no means surprising: In equilibrium, the system tends to minimize a diamagnetic response to the exter- nal field H. Note that for N # �, H � 0, L � �, equa- tions (A1) admit an exact soliton—antisoliton solution " � � "n ny y( ) ( ) ( )1 , " � 0 1 22 3 4 55( ) exp min y y 4 arctan � , (A8) Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 869 � � -� - min [ ]% � � N/ J 2 24 . (A9) However, solution (A8), (A9) vanishes for any H � 0, L � �. Note the first integral of (A1): cos ( ) ( , ) ( ) ( ) . n n J mn n my G n m d y dy d y dy C� ��" � " " � - �2 2 2 (A10) In the case of a finite interval y L L6 �[ , ], the constant of integration C can be determined from the condi- tions (27), (28): C NH H L s n n� � " ��2 2 2 cos ( ). (A11) In the infinite interval y 6 �� ��( , ), C N� . Now, we will prove that Eqs. (83)–(89) provide a complete set of stable solutions to (A1) at H � 0. Using equations of Secs. 2.1, 2.2 with fn % 1and intro- ducing the «local magnetic field» in the regions ( )n p x np� � �1 via h y ep G n m d y dyn m m( ) ( , ) ( ) � "�-2 2 , (A12) we rewrite (3) as follows: � � �LD n n n nd dy H d dy H" " � � � � � " " � � �, ; , ;* 0 , (A13) �* , ; [ cos ( )]" " � � � � � " � � ��n n z L L n s n d dy H pW dy H y 8 2 1 2 � � � � � � � � � �� 1 2 1 2 2 - [ ( ) ( )] [ ( ) ]h y h y h y Hn n n � � " " � � � � � NpWW H d dy H Wz H n n z 2 8 4 0 � � � �* , , (A14) where �H n nd dy * ," " � � � � � " � � �� � �H pW dy ys z L L n n 2 16 1 � [ cos ( )] � " " � � � ���- �2 2 2 0J mn n mG n m d y dy d y dy ( , ) ( ) ( ) (A15) is the Helmholtz free-energy functional, and the total flux is given by � � " � " ��1 2e L Ln n n [ ( ) ( )]. (A16) The treatment of the functionals �*, �H * is fully analogous to that of �, �H in Sec. 2.3. Thus, by vir- tue of positive definitiveness of (A4), the Legendre necessary condition for a strong minimum [19], G n m mn n m n n �� �� <( , ) ,= = =0 02 , (A17) where =n are arbitrary real numbers, is explicitly ful- filled. The functionals �*, �H * have a common set of minimizers. The stationarity condition for �H * in- volves the vanishing of the volume variation, which yields Eqs. (A3), and of the surface variation, which leads, by (27) and (A16), to the conservation law for the flux �� � 0. By analogy with (69), (70), we ar- rive at soliton boundary conditions ' (" � % " � � � � � � � � �n n L L nZ dy d y dy n N ( ) ( ) , ..., 0 1 2 0 1 � � , (A18) � " � % "� � � �� � � !� � � " � � � �n L L n n L dy d y dy L ( ) ( ) , ( ) � � � 1 2 0. (A19) The soliton solution in the sector > ?Z ZN1,..., first appears under N conditions " � � � " � �1 0( ) ... ( )L LN (A20) [see (A19)] and N � 1 conditions d dy L d dy L epHN" � � � " � � �1 2 0( ) ... ( ) * , (A21) where the field H* is so far undetermined [compare with (27), (28)]. Given that the general solution to (A1) contains 2N constants of integration, conditions (A20), (A21) leave only one undetermined constant to satisfy the boundary conditions (A18). For this reason, we have to set Z Z Z NN v1 0 1� � % �... , , ... (A22) For H* � 0, by the Proposition, the unique solution to the initial value problem (A20), (A21) in the in- terval y L L6 �[ , ] is the trivial Meissner configuration " � � " %1 0... N , Nv � 0. For arbitrary H* � 0, by the Proposition, the initial value problem (A20), (A21) also admits a unique solution in the interval y L L6 �[ , ], and its explicit form is " � � " % " �1( ) ... ( ) ( )y y yN � � � � � 0 1 2 2 3 4 5 5� � 2 2am y L k K k k J ( ), , (A23) 870 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky k H H H s s � �*2 2 . (A24) Upon substitution of (A23), (27) into (A18), with (A22) and Nv � 1 2, , ..., we determine H* � � ��H H N s v 1 2 2 , where HNv are given by (41). In this way, we arrive at the solutions (83)–(89), which proves their uniqueness as minimizers of (A13)– (A15). This proof clearly demonstrates that the ab- sence of soliton solutions of any other types is a result of the physical boundary conditions (A21) [or (27), (28)]. To establish a relationship to the exact variational method of Refs. 1, 2, we note that (83)–(89) can also be obtained by the minimization of (A13)–(A15) with respect to the phases � n . (We recall that " �n n% � � �� n 1.) However, as first noticed in Refs. 1, 2, we must take into account that not all � n are independ- ent: The conservation law for the current (22) consti- tutes a constraint on d /dyn� . This problem can be easily circumvented by making use of the first integral (A10), (A11). The substitution of (A10), (A11) into (A15) yields �H n n zL d dy L NWW e p d dy L* ; ( ), ( ) ( )" " � " � � � � � " � � � 32 2� � � 2 � � " � " � � � �H pW dy y Ls z L L n n n 2 16 1 2 � [ cos ( ) cos ( )]. (A25) Taking the variation of (A25) with respect to � n , we immediately arrive at the relations sin ( ) sin ( )" � " �n ny y1 (A26) [compare with (40)], hence the condition (A22) and Eqs. (83)–(89). Appendix B A verification of the Jacobi–Weierstrass–Hilbert sufficient condition for a strong minimum In the weak-coupling limit r T( ) �� 1, when f y( ) % 1, the Legendre necessary condition of the strong mini- mum for �H reduces to the second relation (54). Complemented by the requirement that the explicit solution (83)–(89) can be embedded into a field of extremals, this condition becomes the Jacobi—We- ierstrass—Hilbert sufficient condition for a strong minimum [19]. In view of the symmetry relation " � �" � �( ) ( )y y � 2�Nv and the conditions on variations �" ) �( )L 0, �" �( )0 0, it suffices to verify the Jacobi—Weier- strass—Hilbert condition for y L6 �[ , ]0 . At H � 0, the desired field is given by the one-parameter family " � � �( , ) ( )y Nv; � 1 � � � � 0 1 2 2 3 4 5 5 � 0 1 2 22 3 4 5 55 2 4 2 4 4 2 4 2 2 2 am y K J ; � ; ; , , N m mv � �2 0 1, , , ... ; (B1) " � � 0 1 2 2 3 4 5 5( , ) ,y N y v J ; � ; � ;2 2 2am , N m mv � � �2 1 0 1, , , ... , (B2) where ; � 0. The family "( , )y ; explicitly satisfies (82) and the initial conditions " �( , )0 ; �Nv d dy J " �( , )0 ; ; � . (B3) For ; ;� , where ; � �2 1 2k /k in the case (B1), and ; � 2/k in the case (B2), we have " � "( , ) ( )y y; , i.e., relations (B1) and (B2) yield the solutions (83), (84) and (85), (86), respectively. To prove that the family "( , )y ; indeed forms a field of extremals, we have to show that any two representatives " % "1 1( , )y ; and " % "2 2( , )y ; , where 0 1 2� �; ; , do not intersect in the interval [ , )�L0 . From the first in- tegral of (82) [equation (76) with f % 1] and (B3), we have � ;J Nd dy v 2 1 2 1 1 2 2 2 1 "0 1 22 3 4 55 � " � � �cos ( ) , � ;J Nd dy v 2 2 2 2 2 2 2 2 1 "0 1 22 3 4 55 � " � � �cos ( ) . (B4) We will prove the absence of points of intersection of "1 and "2 by contradiction. Consider ,( ) ( ) ( )y y y% " � "2 1 . As Eq. (82) yields d dy d dy 2 1 2 2 2 2 0 0 0 " � " �( ) ( ) , for y 6 �( , )�1 0 , where �1 is sufficiently small, we have ' (, ; ;( ) | | ( )y y o y� � � �1 2 2 0. By continuity of ,( )y , relation (B5) implies the exis- tence of a finite interval ' (y y6 0 0, where ,( )y � 0. Let y y L� 6 �0 0[ , ) be a point of intersection, i.e., " � " " � " 1 0 2 0 2 1 2 0 2 2 2 0( ) ( ), ( ) ( )y y d dy y d dy y . Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 871 For y y y6 �( , )0 0 � , where � � 0 is sufficiently small, we have ,( ) ( ) ( ) (| | | | ) [( )y d dy y d dy y y y o y y� " � " � � � � � �2 0 1 0 0 0 2] � 0 (B5) which, in view of the arbitrariness of �, implies d dy y d dy y " � "2 0 1 0( ) ( ). (B6) However, equations (B4), by virtue of d dy d dy " " �1 2 0, , yield d dy y d dy y " � "2 0 1 0( ) ( ). (B7) The contradiction between (B6) and (B7) proves the absence of points of intersection in the whole interval [ , )�L0 , as expected. Thus, the solution (83)–(89), for any H � 0, can be embedded into a field of extremals and, as such, satisfies the Jacobi—Weierstrass—Hilbert sufficient condition for a strong minimum. Appendix C A comparison between soliton and non-soliton («lattice») configurations At fields H H eps J�� %� �- - �1 1( ) , (C1) the SG equations (A1) with N m� # �2 admit an ex- act, closed-form analytical non-soliton solution under the conditions d dy L epH nn n " ) � " �( ) , ( )2 0 � . (C2) A solution of this type was proposed, e.g., in Refs. 7, 14, where it was erroneously interpreted as a «dense triangular lattice of Josephson vortices». As an illus- tration of the general results of the paper, it is in- structive to compare this solution with the exact closed-form analytical soliton solution (92), valid in a wider field range H Hs�� . By introducing a dimensionless variable u epHy% 2 and new functions ' (, n nu u/ epH( ) % " 2 , we rewrite Eqs. (A1) as d u du ep H G n m un J m m 2 2 2 11 2 , - � , ( ) ( ) ( , ) sin ( )� � � , n N� 1, ... , . (C3) The boundary conditions, Eqs. (C2), become d du epHL nn n , �( ) , ( )) � " �2 1 0 . (C4) Taking into account (C1), we seek the solution to (C3), (C4) as an asymptotic expansion in powers of 1 2 2/ ep HJ( )� : , ,n n k k u u( ) ( )( )� � � 0 , (C5) where , n k u( )( ) is of order 1 2 2/ ep HJ k( )- � (k � 0 1, , ...). Retaining only the first two terms in (C5), we obtain ' (" � � � � � �n n J y n epHy ep H epHy( ) ( ) ( ) sin� - - � 2 4 1 4 2 2 2 2 � � �4 1 2 2 2 2 - - � ( ) cos ( ) n Jep H epHW y. (C6) The sum of the first three terms on the right hand side of (C6) in the limit - �� 1 gives the solution of Refs. 7, 14. The presence of the last term on the right-hand side of (C6), resulting from the boundary conditions (C2) at y L� ) , was not noticed in Refs. 7, 14, and therefore the solution of Refs. 7, 14 does not meet the boundary conditions at y L� ) in required order. In contrast to the exact closed-form analytical soliton solution (92), valid in the same field range and minimizing the LD functional (3), the non-so- liton solution (C6) is just a saddle point of (3): see Secs. 2.3 and 4 and Appendix A. This is illustrated below. The substitution of (C6) into (A13), (A14) yields a nonequilibrium value of the LD functional: ~ ( ) ( ) �LD c zH H T pNLW � � 2 4� � � � � � � � � � � � � � 1 1 1 2 8 2 2 2 r T / epHW ep HJ ( ) [ ] cos ( ) ( ) - � � � . (C7) Expression (C7) is to be compared with the thermo- dynamic free energy (93) of the soliton solution (92). Their difference is $� � �~ ( ) ~ ( ) ( )LD LD LDH H H% � � � � H T pNLW r Tc z 2 4 ( ) ( ) � � � � � � � � � | sin ( )| cos ( ) ( ) . epHW epHW epHW ep HJ 2 2 24 0 � - (C8) Thus, the saddle-point, non-soliton solution (C6) is nothing but an unstable perturbation of the soliton solution (92). 872 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 Sergey V. Kuplevakhsky 1. S.V. Kuplevakhsky, Phys. Rev. B60, 7496 (1999). 2. S.V. Kuplevakhsky, Phys. Rev. B63, 054508 (2001). 3. W.E. Lawrence and S. Doniach, in: Proceedings of the Twelfth Conference on Low Temperature Physics, Kyoto, 1970, E. Kanda (ed.), Keigaku, Tokyo (1970), p. 361. 4. L.N. Bulaevskii, Zh. Eksp. Teor. Fiz. 64, 2241 (1973) [Sov. Phys. JETP 37, 1133 (1973)]. 5. J.R. Clem and M.W. Coffey, Phys. Rev. B42, 6209 (1990). 6. J.P. Carton, J. Phys. I (France) 1, 113 (1991). 7. L.N. Bulaevskii and J.R. Clem, Phys. Rev. B44, 10234 (1991). 8. L.N. Bulaevskii, M. Ledvij, and V.G. Kogan, Phys. Rev. B46, 366 (1992). 9. A.E. Koshelev, Phys. Rev. B48, 1180 (1993). 10. R. Rajaraman, Solitons and Instantons, North-Hol- land, Amsterdam (1982). 11. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London (1982). 12. P. Goddard and P. Mansfield, Rep. Prog. Phys. 49, 725 (1986). 13. A.S. Schwarz, Quantum Field Theory and Topology Nauka, Moscow (1989) (in Russian). 14. L.N. Bulaevskii, M. Maley, H. Safar, and D. Do- minguez, Phys. Rev. B53, 6634 (1996). 15. L.N. Bulaevskii, J.R. Clem, and L.I. Glazman, Phys. Rev. B46, 350 (1992) 16. V.M. Krasnov, Phys. Rev. B63, 064519 (2001). 17. V.M. Krasnov, Phys. Rev. B65, 096503 (2002). 18. C. Lancsos, The Variational Principles of Mechanics, University of Toronto Press, Toronto (1962). 19. N.I. Akhiezer, The Calculus of Variations, Blaisdell Publishing, New York (1962). 20. N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York (1976). 21. A.O. Gelfond, The Calculus of Finite Difference, GIFML, Moscow (1959) (in Russian). 22. A.A. Abrikosov, Fundamentals of the Theory of Me- tals, North-Holland, Amsterdam (1988). 23. V.E. Zakharov and E.A. Kuznetsov, Zh. Eksp. Teor. Fiz. 66, 594 (1974). 24. N.D. Mermin, Rev. Mod. Phys. 51, 591 (1979). 25. M. Abramowitz and I.A. Stegun, Handbook of Ma- thematical Functions, Dover, New York (1965). 26. I.O. Kulik, Zh. Eksp. Teor. Fiz. 51, 1953 (1966) [Sov. Phys. JETP 24, 1307 (1967)]. 27. C.S. Owen and D.J. Scalapino, Phys. Rev. 164, 538 (1967). 28. A. Barone and G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York (1982). 29. A. Buzdin and D. Feinberg, Phys. Lett. A165, 281 (1992). 30. B. Farid, J. Phys. Condens. Matter 10, L589 (1998)]. 31. R. Bellman, Introduction to Matrix Analysis, McGraw- Hill, New York (1960). 32. V.C.L. Hutson and J.S. Pym, Applications of Func- tional Analysis and Operator Theory, Academic Press, London (1980). Topological solitons of the Lawrence–Doniach model as equilibrium Josephson vortices Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 873

Ähnliche Einträge