Explicit Solutions for a Nonlinear Vector Model on the Triangular Lattice
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2019), 028, 17 pages Explicit Solutions for a Nonlinear Vector Modelon the Triangular Lattice
V.E. VEKSLERCHIKUsikov Institute for Radiophysics and Electronics, 12 Proskura Str., Kharkiv, 61085, Ukraine
E-mail: [email protected]
Received December 28, 2018, in final form April 04, 2019; Published online April 13, 2019https://doi.org/10.3842/SIGMA.2019.028
Abstract.
We present a family of explicit solutions for a nonlinear classical vector modelwith anisotropic Heisenberg-like interaction on the triangular lattice.
Key words: classical Heisenberg-type models; triangular lattice; bilinear approach; explicitsolutions; solitons
In this paper, which is a continuation of [21, 22], we study the possibilities to construct explicitsolutions for nonlinear lattices with non-square geometry. Our aim is twofold. First, we want topresent an application of some of the ideas and methods elaborated for models on such latticesand even on arbitrary graphs (see, for example, [2, 3, 4, 5, 6, 7, 8, 11]). On the other hand,we want to demonstrate that some of the results obtained for the rectangular lattices can bemodified to provide solutions for the models defined on the triangular ones.We consider a nonlinear vector model defined on the triangular lattice with interaction be-tween the nearest neighbours, S = (cid:88) n . n . L ( (cid:126)n , (cid:126)n ) (1.1)described by L ( (cid:126)n , (cid:126)n ) = J (cid:126)n ,(cid:126)n ln (cid:12)(cid:12) K (cid:126)n ,(cid:126)n (cid:104) (cid:126)q ( (cid:126)n ) − (cid:126)q ( (cid:126)n ) , (cid:126)r ( (cid:126)n ) − (cid:126)r ( (cid:126)n ) (cid:105) (cid:12)(cid:12) (1.2)and the restriction (cid:104) (cid:126)q ( (cid:126)n ) , (cid:126)r ( (cid:126)n ) (cid:105) = 1 (1.3)(for all (cid:126)n ) where the brackets (cid:104) , (cid:105) stand for the standard scalar product.From the physical viewpoint, the model (1.1) with the Lagrangian (1.2), which can be rewrit-ten, due to the restriction (1.3), as L ( (cid:126)n , (cid:126)n ) = J (cid:126)n ,(cid:126)n ln (cid:12)(cid:12) K (cid:48) (cid:126)n ,(cid:126)n (cid:104) (cid:126)q ( (cid:126)n ) , (cid:126)r ( (cid:126)n ) (cid:105) + K (cid:48) (cid:126)n ,(cid:126)n (cid:104) (cid:126)q ( (cid:126)n ) , (cid:126)r ( (cid:126)n ) (cid:105) (cid:12)(cid:12) + const , is one of the classical versions of the famous anisotropic Heisenberg spin model of the quantummechanics [12, 14, 16].Comparing this work with the previous one, [22], it should be noted that here we studythe anisotropic interaction (in [22] the interaction was with K (cid:126)n ,(cid:126)n = 1) on the triangularlattice (instead of the honeycomb one) and take into account the restriction (1.3), which is veryimportant for possible physical applications. a r X i v : . [ n li n . S I] A p r V.E. VekslerchikIn the next section we discuss in more detail the Lagrangian (1.2) and the equations which arethe main subject of our study. In Section 3, we introduce some auxiliary system and demonstratehow it can used to derive solutions for our equations. These results are used in Section 4 topresent three families of the explicit solutions: two types of the N -soliton solutions and onesconstructed of the determinants of the Toeplitz matrices. Finally, in the conclusion we discussthe limitations of the method we use in this paper and possible continuations of the presentedstudies. It should be noted that, although we study a two-dimensional lattice, we introduce, instead ofa pair of basis vectors, three coplanar vectors (cid:126)e , (cid:126)e and (cid:126)e related by (cid:80) i =1 (cid:126)e i = (cid:126) (cid:126)n as a linear combinations (cid:126)n = (cid:80) i =1 n i (cid:126)e i (see Fig. 1).In terms of { (cid:126)e i } , the set of the nearest neighbours of a lattice point (cid:126)n is given by { (cid:126)n ± (cid:126)e i } i =1 , , and the discrete action (1.1) can be rewritten as S = 12 (cid:88) (cid:126)n (cid:88) i =1 (cid:88) ε = ± L ( (cid:126)n, (cid:126)n + ε(cid:126)e i ) . (2.1)We cannot solve the most general variant of this model and hereafter impose some restrictionson the coefficients J (cid:126)n ,(cid:126)n and K (cid:126)n ,(cid:126)n in (1.2). First we assume that our model is homogeneous:the interaction depends only on the direction of (cid:126)n − (cid:126)n , J (cid:126)n,(cid:126)n ± (cid:126)e i = J i , K (cid:126)n,(cid:126)n ± (cid:126)e i = K i . Secondly, we assume that constants J i are related by (cid:88) i =1 J i = 0 (2.2)and that K i = 14 sinh κ i , (cid:88) i =1 κ i = 0 . (2.3)Alternatively, we can formulate these restrictions as J i = (cid:104) (cid:126)J , (cid:126)e i (cid:105) , K i = 14 sinh (cid:104) (cid:126)K, (cid:126)e i (cid:105) (2.4)with arbitrary vectors (cid:126)J and (cid:126)K . The restriction (2.2) is common for various integrable models.Considering the second restriction, (2.3), we admit that is has no clear physical explanation andis introduced to make equations (2.5) and (2.6) solvable by the method discussed in what followsrather than to ensure appearance of some additional symmetry of the model or its integrability.Finally, the big part of the calculations presented below is valid for vectors (cid:126)q, (cid:126)r ∈ R n or (cid:126)q, (cid:126)r ∈ C n with arbitrary n . However, as it will be shown below, there arise some problems that,at present, we can solve only in the n = 2 case. So, we state from the beginning that we consideronly two-dimensional real vectors (cid:126)q and (cid:126)r , (cid:126)q, (cid:126)r ∈ R . xplicit Solutions for a Nonlinear Vector Model on the Triangular Lattice 3 J , K J , K J , K J ,K J , K J , K ⃗e ⃗e ⃗e Figure 1.
The triangular lattice (left), the constants (center) and the base vectors (cid:126)e i (right). Looking for the extremals of the action (2.1) under the constraints (1.3) one arrives at thefield equations of our model, (cid:88) i =1 J i K i (cid:26) (cid:126)q ( (cid:126)n + (cid:126)e i ) − (cid:104) (cid:126)q ( (cid:126)n + (cid:126)e i ) , (cid:126)r ( (cid:126)n ) (cid:105) (cid:126)q ( (cid:126)n )1 + K i (cid:104) (cid:126)q ( (cid:126)n ) − (cid:126)q ( (cid:126)n + (cid:126)e i ) , (cid:126)r ( (cid:126)n ) − (cid:126)r ( (cid:126)n + (cid:126)e i ) (cid:105) + (cid:126)q ( (cid:126)n − (cid:126)e i ) − (cid:104) (cid:126)q ( (cid:126)n − (cid:126)e i ) , (cid:126)r ( (cid:126)n ) (cid:105) (cid:126)q ( (cid:126)n )1 + K i (cid:104) (cid:126)q ( (cid:126)n ) − (cid:126)q ( (cid:126)n − (cid:126)e i ) , (cid:126)r ( (cid:126)n ) − (cid:126)r ( (cid:126)n − (cid:126)e i ) (cid:105) (cid:27) = 0 , (2.5) (cid:88) i =1 J i K i (cid:26) (cid:126)r ( (cid:126)n + (cid:126)e i ) − (cid:104) (cid:126)q ( (cid:126)n ) , (cid:126)r ( (cid:126)n + (cid:126)e i ) (cid:105) (cid:126)r ( (cid:126)n )1 + K i (cid:104) (cid:126)q ( (cid:126)n ) − (cid:126)q ( (cid:126)n + (cid:126)e i ) , (cid:126)r ( (cid:126)n ) − (cid:126)r ( (cid:126)n + (cid:126)e i ) (cid:105) + (cid:126)r ( (cid:126)n − (cid:126)e i ) − (cid:104) (cid:126)q ( (cid:126)n ) , (cid:126)r ( (cid:126)n − (cid:126)e i ) (cid:105) (cid:126)r ( (cid:126)n )1 + K i (cid:104) (cid:126)q ( (cid:126)n ) − (cid:126)q ( (cid:126)n − (cid:126)e i ) , (cid:126)r ( (cid:126)n ) − (cid:126)r ( (cid:126)n − (cid:126)e i ) (cid:105) (cid:27) = 0 . (2.6)Namely these equations are the main subject of this work. In the following sections we presentthree families of explicit solutions for the system (2.5) and (2.6). The ansatz that we use in what follows is closely related to the already known ideas that maybe termed ‘star-triangle’ (or ‘star-polygon’) transformation or, using the language of, e.g., [7],the ‘three-leg’ representation.We do not start with the form of solutions, which is usual way to introduce an ansatz. Instead,we demonstrate that a wide range of solutions for (2.5) and (2.6) can be obtained from a moresimple system of equations.
The main steps of the proposed ansatz can be described as follows. First, we consider the originallattice { (cid:126)n } together with its dual, { (cid:126)n ± (cid:126)g i } , where new vectors (cid:126)g i are related to (cid:126)e i by (cid:126)e i = (cid:126)g i +1 − (cid:126)g i − . (3.1)In this equation, as well as in the rest of the paper, we use the following convention: all arithmeticoperations with the indices of the vectors (cid:126)e i and (cid:126)g i and the constants J i , K i and other areunderstood modulo 3, (cid:126)e i ± = (cid:126)e i , (cid:126)g i ± = (cid:126)g i , J i ± = J i , K i ± = K i , etc , i = 1 , , . In other words, we write i ± i + 1 := i = 1 , i = 2 , i = 3 , i − i = 1 , i = 2 , i = 3 . V.E. Vekslerchik ⃗e ⃗e ⃗e ⃗g ⃗g ⃗g I , H I , H I , H Figure 2.
The base vectors (cid:126)g i (left), and the new constants (right). Each rhombus in the right figurerepresents the set of nodes involved in the system (3.5). Then, we introduce a rather simple system of four-point equations relating the values of (cid:126)q and (cid:126)r at the points (cid:126)n , (cid:126)n + (cid:126)g i and (cid:126)n + (cid:126)g j + (cid:126)g k and demonstrate that each solution of the latter isat the same time a solution of the field equations (2.5) and (2.6). This system can be written as H i − (cid:126)q ( (cid:126)n + (cid:126)g i − ) − H i +1 (cid:126)q ( (cid:126)n + (cid:126)g i +1 ) = 1 J i F i ( (cid:126)n ) (cid:126)q ( (cid:126)n + (cid:126)g i − + (cid:126)g i +1 ) ,H i − (cid:126)r ( (cid:126)n + (cid:126)g i +1 ) − H i +1 (cid:126)r ( (cid:126)n + (cid:126)g i − ) = 1 J i F i ( (cid:126)n ) (cid:126)r ( (cid:126)n ) (3.2)with constant H i ( i = 1 , ,
3) and F i ( (cid:126)n ) = 1 K i + (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i +1 ) − (cid:126)q ( (cid:126)n + (cid:126)g i − ) , (cid:126)r ( (cid:126)n + (cid:126)g i +1 ) − (cid:126)r ( (cid:126)n + (cid:126)g i − ) (cid:105) (note that structure of F i ( (cid:126)n ) is that of the denominators of the summands in (2.5) and (2.6)).It turns out that the consistency of the system (3.2) implies the following restrictions on theconstants J i : J i = I i +1 − I i − , (3.3)where I i are arbitrary constants and the following relations between K i and H i : K i = H i − H i +1 ( H i − − H i +1 ) (3.4)(see Appendix A). Finally, we arrive at the main result of this paper which can be formulatedas follows. Proposition 3.1.
Each solution of the system H i − (cid:126)q ( (cid:126)n + (cid:126)g i − ) − H i +1 (cid:126)q ( (cid:126)n + (cid:126)g i +1 ) = 1 I i +1 − I i − F i ( (cid:126)n ) (cid:126)q ( (cid:126)n + (cid:126)g i − + (cid:126)g i +1 ) ,H i − (cid:126)r ( (cid:126)n + (cid:126)g i +1 ) − H i +1 (cid:126)r ( (cid:126)n + (cid:126)g i − ) = 1 I i +1 − I i − F i ( (cid:126)n ) (cid:126)r ( (cid:126)n ) (3.5) with F i ( (cid:126)n ) = ( H i − − H i +1 ) H i − H i +1 + (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i +1 ) − (cid:126)q ( (cid:126)n + (cid:126)g i − ) , (cid:126)r ( (cid:126)n + (cid:126)g i +1 ) − (cid:126)r ( (cid:126)n + (cid:126)g i − ) (cid:105) (3.6) solves equations (2.5) and (2.6) with the constants J i , K i and the vectors (cid:126)e i given by (3.3) , (3.4) and (3.1) . xplicit Solutions for a Nonlinear Vector Model on the Triangular Lattice 5We would like to repeat that this proposition describes an ansatz (or reduction): each solutionfor (3.5) with (3.6) satisfies equations (2.5) and (2.6) but the reverse statement is surely not true,only a part of solutions for (2.5) and (2.6) can be obtained from the system (3.5) with (3.6).The proof of Proposition 3.1 is rather simple except of one non-trivial fact following from theconsistency of the system (3.5): the scalar products (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i ) , (cid:126)r ( (cid:126)n ) (cid:105) do not depend on (cid:126)n (weprove this statement in Appendix A).After this fact is established, the rest of calculations is easy. To compute the summandin (2.5), (cid:126)Q ± i ( (cid:126)n ) = J i f ± i ( (cid:126)n ) { (cid:126)q ( (cid:126)n ± (cid:126)e i ) − (cid:104) (cid:126)q ( (cid:126)n ± (cid:126)e i ) , (cid:126)r ( (cid:126)n ) (cid:105) (cid:126)q ( (cid:126)n ) } , where f ± i ( (cid:126)n ) = 1 K i + (cid:104) (cid:126)q ( (cid:126)n ) − (cid:126)q ( (cid:126)n ± (cid:126)e i ) , (cid:126)r ( (cid:126)n ) − (cid:126)r ( (cid:126)n ± (cid:126)e i ) (cid:105) we express (cid:126)q ( (cid:126)n ± (cid:126)e i ) = (cid:126)q ( (cid:126)n ± (cid:126)g i +1 ∓ (cid:126)g i − ) and (cid:104) (cid:126)q ( (cid:126)n ± (cid:126)e i ) , (cid:126)r ( (cid:126)n ) (cid:105) from the first equation from(3.5) (translated by − (cid:126)g i ∓ ), note that f ± i ( (cid:126)n ) = F ( (cid:126)n − (cid:126)g i ∓ ), impose the restriction (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i ) , (cid:126)r ( (cid:126)n ) (cid:105) = H i I i (3.7)and arrive at (cid:126)Q i ( (cid:126)n ) = − (cid:126)Q (cid:48) i +1 ( (cid:126)n ) , (cid:126)Q − i ( (cid:126)n ) = (cid:126)Q (cid:48) i − ( (cid:126)n ) , i = 1 , , , where (cid:126)Q (cid:48) i ( (cid:126)n ) = 1 H i (cid:126)q ( (cid:126)n + (cid:126)g i ) − I i (cid:126)q ( (cid:126)n ) . Now, one can easily obtain (cid:88) i =1 (cid:126)Q i ( (cid:126)n ) = − (cid:88) i =1 (cid:126)Q (cid:48) i ( (cid:126)n ) , (cid:88) i =1 (cid:126)Q − i ( (cid:126)n ) = (cid:88) i =1 (cid:126)Q (cid:48) i ( (cid:126)n ) , which means that the right-hand side of (2.5) vanishes automatically,r.h.s. (2.5) = (cid:88) i =1 (cid:2) (cid:126)Q i ( (cid:126)n ) + (cid:126)Q − i ( (cid:126)n ) (cid:3) = 0 . Equation (2.6) can be treated in the similar way.
One of the ingredients of the proposed method to derive solutions for the field equations (2.5)and (2.6) is to use, instead the vectors (cid:126)e i , the vectors (cid:126)g i . It is natural to think of the latter aspointing to the centers of the (three of six) triangular plaquettes adjacent to the point (cid:126)n = (cid:126) (cid:126)g i with this geometrical meaning. One canconsider them just as three vectors satisfying (3.1) for given set of (cid:126)e i . It is easy to show thatequation (3.1) admits one-parametric family of solutions, (cid:126)g i = (cid:126)g ∗ + ( (cid:126)e i − − (cid:126)e i +1 ) (3.8) V.E. Vekslerchikwith arbitrary (cid:126)g ∗ . The fact that the system (3.1) does not determine (cid:126)g i uniquely is not importantfor our purposes: in the worst case we may have different solutions for different choices of (cid:126)g ∗ .However, as we see in what follows, this arbitrariness does not affect the final formulae forsolutions.In a similar way, relations (3.3) and (3.4) considered as equations from which one has tofind I i and H i lead to I i = I ∗ + ( J i − − J i +1 ) (3.9)and H i = H ∗ exp (cid:2) ( κ i − − κ i +1 ) (cid:3) , (3.10)where I ∗ and H ∗ are arbitrary constants or, if one uses (2.4), I i = I (cid:48)∗ + (cid:104) (cid:126)J , (cid:126)g i (cid:105) , H i = H (cid:48)∗ exp (cid:2) (cid:104) (cid:126)K, (cid:126)g i (cid:105) (cid:3) with arbitrary I (cid:48)∗ and H (cid:48)∗ .It should be noted that when we use the three vectors (cid:126)e i to describe the triangular lattice,instead of a two-vector basis, we bear in mind the restriction (cid:80) i =1 (cid:126)e i = 0. This restriction leads tosome technical problems when constructing the explicit solutions. Now, if we do not introducethe vectors (cid:126)g i as pointing to the centers of the plaquettes, but use (3.8) as the definition, wedo not have the ‘geometric’ restriction: (cid:80) i =1 (cid:126)g i (cid:54) = 0. Thus, one of advantages of the proposedconstruction is that it solves the question of the (cid:80) i =1 (cid:126)e i = 0 constraint. Here, we derive explicit solutions for the ansatz equations (3.5) and (3.6). To simplify thefollowing formulae and to make the correspondence with the previous works we change thenotation. First, we introduce the shifts T to indicate the translations: T g i (cid:126)q ( (cid:126)n ) = (cid:126)q ( (cid:126)n + (cid:126)g i ) . Then, we rewrite the system of the three equations (3.5) as difference/functional equationsrelating (cid:126)q and (cid:126)r with T ξ (cid:126)q , T ξ (cid:126)r , T η (cid:126)q and T η (cid:126)r where ξ and η belong to the set of parameters { g , g , g } that correspond to the translations by the vectors { (cid:126)g , (cid:126)g , (cid:126)g } . This can be doneby redefinition of the constants: I i → ˙ I g i , H i → ˙ H g i . In new terms, equations (3.5) and (3.6)become˙ H ξ T ξ (cid:126)q − ˙ H η T η (cid:126)q = 1˙ I η − ˙ I ξ F ξ,η T ξη (cid:126)q, ˙ H ξ T η (cid:126)r − ˙ H η T ξ (cid:126)r = 1˙ I η − ˙ I ξ F ξ,η (cid:126)r (4.1)and F ξ,η = ˙ H ξ ˙ H η + ˙ H η ˙ H ξ − (cid:104) T ξ (cid:126)q, T η (cid:126)r (cid:105) − (cid:104) T η (cid:126)q, T ξ (cid:126)r (cid:105) , (4.2)xplicit Solutions for a Nonlinear Vector Model on the Triangular Lattice 7and can be easily bilinearized by the substitution (cid:126)q = (cid:126)στ , (cid:126)r = (cid:126)ρτ and F ξ,η = B ξ,η τ T ξη τ (cid:0) T ξ τ (cid:1)(cid:0) T η τ (cid:1) with arbitrary constants B ξ,η , symmetric in ξ and η ( B ξ,η = B η,ξ ), which leads to the bilinearsystem˙ H ξ (cid:0) T η τ (cid:1) T ξ (cid:126)σ − ˙ H η (cid:0) T ξ τ (cid:1) T η (cid:126)σ = A ξ,η τ T ξη (cid:126)σ, ˙ H ξ (cid:0) T ξ τ (cid:1) T η (cid:126)ρ − ˙ H η (cid:0) T η τ (cid:1) T ξ (cid:126)ρ = A ξ,η (cid:0) T ξη τ (cid:1) (cid:126)ρ (4.3)and B ξ,η τ T ξη τ + C ξ,η (cid:0) T ξ τ (cid:1)(cid:0) T η τ (cid:1) = (cid:104) T ξ (cid:126)σ, T η (cid:126)ρ (cid:105) + (cid:104) T η (cid:126)σ, T ξ (cid:126)ρ (cid:105) , (4.4)where A ξ,η = B ξ,η ˙ I η − ˙ I ξ , C ξ,η = ˙ H ξ ˙ H η + ˙ H η ˙ H ξ . (4.5)It should be noted that equation (4.4) turns out to be the direct consequence of the equa-tions (4.3), the definitions (3.3) and (3.4) together with (1.3) and (3.7) which can be rewrittenas (cid:104) (cid:126)σ, (cid:126)ρ (cid:105) = τ , (cid:104) T ξ (cid:126)σ, (cid:126)ρ (cid:105) = ˙ H ξ ˙ I ξ τ (cid:0) T ξ τ (cid:1) . These equations are, in some sense, an alternative to (4.4) and in the next sections, we willderive solutions for (4.1)–(4.2), and hence for (3.5)–(3.6), using the following
Proposition 4.1.
Any solution for the system ˙ H ξ (cid:0) T η τ (cid:1) T ξ (cid:126)σ − ˙ H η (cid:0) T ξ τ (cid:1) T η (cid:126)σ = A ξ,η τ T ξη (cid:126)σ, (4.6)˙ H ξ (cid:0) T ξ τ (cid:1) T η (cid:126)ρ − ˙ H η (cid:0) T η τ (cid:1) T ξ (cid:126)ρ = A ξ,η (cid:0) T ξη τ (cid:1) (cid:126)ρ, (4.7) (cid:104) (cid:126)σ, (cid:126)ρ (cid:105) = τ , (4.8) (cid:104) T ξ (cid:126)σ, (cid:126)ρ (cid:105) = ˙ H ξ ˙ I ξ τ (cid:0) T ξ τ (cid:1) , (4.9) with arbitrary skew-symmetric constants A ξ,η ( A ξ,η = − A η,ξ ) provides solution for the sys-tem (4.1) – (4.2) with constants B ξ,η and C ξ,η given by (4.5) by means of (cid:126)q = (cid:126)στ , (cid:126)r = (cid:126)ρτ . Equations (4.6) and (4.7) are known for many years. In this form, or similar ones, theyhave appeared in the studies of various integrable systems. For example, the compatibility of,say, equations (4.6) implies that τ should solve the famous Hirota bilinear difference equation(HBDE) [13], also known as the discrete KP equation, A ξ,η ˙ H ζ (cid:0) T ξη τ (cid:1)(cid:0) T ζ τ (cid:1) − A ξ,ζ ˙ H η (cid:0) T ξζ τ (cid:1)(cid:0) T η τ (cid:1) + A η,ζ ˙ H ξ (cid:0) T ηζ τ (cid:1)(cid:0) T ξ τ (cid:1) = 0 . (4.10) V.E. VekslerchikThus, equations (4.6) (or (4.7)) can be viewed as the Lax representation of the HBDE (see, e.g.,[9, 10, 15, 17, 23]). Moreover, it turns out that all components of (cid:126)σ and (cid:126)ρ also solve (4.10),thus equations (4.6) and (4.7) can be interpreted as describing the B¨acklund transformationsfor the HBDE. Also, system (4.6) and (4.7) is closely related to another integrable model: it de-scribes the so-called functional representation of the Ablowitz–Ladik hierarchy [1] (compare (4.6)and (4.7) with equation (6.10) of [18]).Considering the remaining equations of the auxiliary system, (4.8) and (4.9) (or equa-tion (4.4)), they can be viewed, in the framework of the theory of the HBDE, as a nonlinearrestriction, which is compatible with (4.6) and (4.7). And namely this is the point that makespresented work different from, say, [22]: there we used another way to ‘close’ equations (4.6)and (4.7), i.e., to relate (cid:126)σ and (cid:126)ρ with the tau-function τ .Returning to our current task, we would like to repeat that equations (4.6)–(4.9) are not newand one does not need to derive some special methods to find their solutions. In what follows wepresent three families of explicit solutions for (4.6)–(4.9), and, hence, for (4.1) and (4.2). In allthe cases our approach is to establish the relations between equations we want to solve and thealready known equations like the Jacobi identities for the Toeplitz matrices or various identitiesfor the Cauchy-type matrices which gives us a possibility to obtain solutions by means of rathersimple calculations.One of the differences between these cases is different dependence of the constants ˙ I ξ and ˙ H ξ on ξ . To return from the solutions for (4.6)–(4.9) to solutions for the field equations (2.5)and (2.6) we have to invert this dependence: we have to express ξ in terms of ˙ I ξ and ˙ H ξ , i.e.,to obtain the dependence of g i on ˙ I g i and ˙ H g i . Then, recalling that ˙ H g i = H i and ˙ I g i = I i are functions of J i and K i given by (3.9), (3.10) and (2.3), we establish how the parameters g i depend on J i and K i . The solutions we discuss in this section are built of the determinants of the Toeplitz matrices∆ MN = det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α M α M +1 . . . α M + N − α M − α M . . . α M + N − ... ... . . . ... α M − N +1 α M − N +2 . . . α M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.11)and the shifts defined by T − ξ α m = α m − ξα m +1 . The action of the shifts can be ‘translated’ to the level of determinants ∆ MN as T − ξ ∆ MN = ( − ) N det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ N ξ N − . . . α M α M +1 . . . α M + N ... ... . . . ... α M − N +1 α M − N +2 . . . α M +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , T − ξη ∆ MN = 1 ξ − η det (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ N +1 ξ N . . . η N +1 η N . . . α M α M +1 . . . α M + N +1 ... ... . . . ... α M − N +1 α M − N +2 . . . α M +2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , etc.xplicit Solutions for a Nonlinear Vector Model on the Triangular Lattice 9Application of the Jacobi identities to the determinants in the above formulae leads to variousbilinear identities: the ‘basic’ one, (cid:0) ∆ MN (cid:1) = ∆ M − N ∆ M +1 N + ∆ MN − ∆ MN +1 . (4.12)the one-shift,0 = ∆ MN +1 T ξ ∆ MN − ∆ MN T ξ ∆ MN +1 + ξ ∆ M − N T ξ ∆ M +1 N +1 (4.13)or 0 = ∆ M +1 N T ξ ∆ MN − ∆ MN T ξ ∆ M +1 N − ξ ∆ MN − T ξ ∆ M +1 N +1 , (4.14)the two-shift,0 = ( ξ − η )∆ MN T ξη ∆ MN +1 − ξ (cid:0) T ξ ∆ MN +1 (cid:1)(cid:0) T η ∆ MN (cid:1) + η (cid:0) T ξ ∆ MN (cid:1)(cid:0) T η ∆ MN +1 (cid:1) , (4.15)0 = ( ξ − η )∆ MN T ξη ∆ M +1 N − ξ (cid:0) T ξ ∆ M +1 N (cid:1)(cid:0) T η ∆ MN (cid:1) + η (cid:0) T ξ ∆ MN (cid:1)(cid:0) T η ∆ M +1 N (cid:1) , (4.16)etc. These identities are the simplest ones, from which one can deduce an infinite set of bilinearidentities for the Toeplitz determinants.Using equations (4.12)–(4.16) and their direct consequences it is easy to demonstrate thatfunctions τ = ∆ MN , (cid:126)σ = E (cid:32) ∆ MN +1 ∆ M +1 N (cid:33) , (cid:126)ρ = E − (cid:32) ∆ MN − ∆ M − N (cid:33) , where E is the discrete exponential function defined by T ξ E = ˙ H ξ ˙ I ξ E satisfy equations (4.6) and (4.7) with A ξ,η = ξ − η ˙ H ξ ˙ H η ˙ I ξ ˙ I η as well as equations (4.8) and (4.9) provided ζ = ˙ H ζ ˙ I ζ , which means that functions (cid:126)q = (cid:126)σ/τ and (cid:126)r = (cid:126)ρ/τ are solutions for (4.1) and (4.2). Thus,to finish the derivation of the solutions for the field equation the only thing we have to do isto rewrite the above formulae in terms of (cid:126)n bearing in mind that for any function f , f ( (cid:126)n ) = (cid:81) i =1 (cid:0) T g i +1 T − g i − (cid:1) n i f ( (cid:126)
0) and that in our case g i = H i I i with constants I i and H i being definedin (3.9) and (3.10).The (cid:126)n -dependence of the elements of the determinant (4.11) can be then presented by meansof an analogue of the discrete Fourier transform as α m ( (cid:126)n ) = L (cid:88) (cid:96) =1 h m(cid:96) ˆ α (cid:96) ( (cid:126)n )with ˆ α (cid:96) ( (cid:126)n ) = ˆ α (cid:96) (cid:89) i =1 (1 − g i h (cid:96) ) n i +1 − n i − , where ˆ α (cid:96) and h (cid:96) ( (cid:96) = 1 , . . . , L ) are arbitrary constants.0 V.E. VekslerchikTo make the following formulae more clear, we rewrite functions similar to E ( (cid:126)n ) or ˆ α ( h, (cid:126)n )in the exponential form using the following observation: if some function f satisfies T g i f = C i f with positive constants C i then its (cid:126)n -dependence can be written as f ( (cid:126)n ) = (cid:89) i =1 (cid:0) T g i +1 T − g i − (cid:1) n i f ( (cid:126)
0) = (cid:89) i =1 ( C i +1 /C i − ) n i f ( (cid:126)
0) = f ( (cid:126) e (cid:104) (cid:126)φ, (cid:126)n (cid:105) with (cid:126)φ = 23 (cid:88) i =1 (cid:18) ln C i +1 C i − (cid:19) (cid:126)e i . Gathering the above formulae and making some trivial calculations we can formulate themain result of this section as follows.
Proposition 4.2.
The N th order Toeplitz solutions for the field equations can be presented as (cid:126)q ( (cid:126)n ) = C e (cid:104) (cid:126)ϕ, (cid:126)n (cid:105) ∆ N ( (cid:126)n ) (cid:32) ∆ N +1 ( (cid:126)n )∆ N ( (cid:126)n ) (cid:33) , (cid:126)r ( (cid:126)n ) = C − e −(cid:104) (cid:126)ϕ, (cid:126)n (cid:105) ∆ N ( (cid:126)n ) (cid:32) ∆ N − ( (cid:126)n )∆ − N ( (cid:126)n ) (cid:33) where ∆ MN ( (cid:126)n ) ( M = 0 , ± is the determinant of the Toeplitz matrix ∆ MN ( (cid:126)n ) = det (cid:12)(cid:12) α M − j + k ( (cid:126)n ) (cid:12)(cid:12) Nj,k =1 , the vector (cid:126)ϕ is given by (cid:126)ϕ = 23 (cid:88) i =1 (cid:18) ln H i +1 I i +1 H i − I i − (cid:19) (cid:126)e i and the function α m ( (cid:126)n ) is defined as α m ( (cid:126)n ) = L (cid:88) (cid:96) =1 ˆ α (cid:96) h m(cid:96) e (cid:104) (cid:126)φ (cid:96) , (cid:126)n (cid:105) with (cid:126)φ (cid:96) = 23 (cid:88) i =1 (cid:18) ln 1 − H i − I i − h (cid:96) − H i +1 I i +1 h (cid:96) (cid:19) (cid:126)e i . Here, ˆ α (cid:96) and h (cid:96) ( (cid:96) = 1 , . . . , L ) , C as well as H ∗ and I ∗ in the definitions (3.10) and (3.9) of H i and I i ( i = 1 , , are arbitrary constants. Here we use some of the results of [19] where we have studied the determinants of the so-called‘dark-soliton’ matrices τ = det | + A | defined by LA − AL − = | (cid:105)(cid:104) a | , xplicit Solutions for a Nonlinear Vector Model on the Triangular Lattice 11where L = diag( L , . . . , L N ), | (cid:105) is the N -column with all components equal to 1, (cid:104) a | is a N -component row that depends on the coordinates describing the model, and their transformationproperties with respect to the shifts defined as T ξ Ω = det | + T ξ A | with T ξ A = AT ξ and the matrices T given by T ξ = ( ξ − L ) (cid:0) ξ − L − (cid:1) − . For our current purposes we need the two facts. First, the determinants τ satisfy the Fay-likeidentity( ξ − η ) (cid:0) T ξη τ (cid:1)(cid:0) T ζ τ (cid:1) + ( η − ζ ) (cid:0) T ηζ τ (cid:1)(cid:0) T ξ τ (cid:1) + ( ζ − ξ ) (cid:0) T ζξ τ (cid:1)(cid:0) T η τ (cid:1) = 0 (4.17)(see [19]) and, secondly, the superposition of T ξ and T /ξ is again the shift, corresponding tozero value of the parameter, T ξ T /ξ = T , (4.18)which follows from the corresponding property of the matrices T ξ , T ξ T /ξ = T , that can beverified straightforwardly.Using only (4.17) and (4.18), without additional referring to the structure of the matrices A ,one can demonstrate that vector-functions (cid:126)σ = (cid:32) q E T − ν τq E T − ν τ (cid:33) , (cid:126)ρ = (cid:32) r E − T ν τr E − T ν τ (cid:33) , where ν , , q , and r , are constants related by ν ν = 1 , q a r a = ν a ν a − , a = 1 , , (4.19)and E , are two discrete exponential functions, defined by T ξ E a = 1˙ H ξ ( ξ − ν a ) E a , a = 1 , , satisfy equations (4.6) and (4.7) with A ξ,η = ( η − ξ ) ˙ H ξ ˙ H η as well as equations (4.8) and (4.9) provided˙ H ξ ˙ I ξ = 1 ξ + ξ − − ν − ν . Thus, to obtain solutions for the equations we are to solve, we have just to return to the (cid:126)n -dependence knowing the action of T ξ . To this end, we have to take ξ ∈ { g , g , g } where g i (parameters that, recall, correspond to the vectors (cid:126)g i ) should be determined from g i + g − i = ν + ν + 1 H i I i (4.20)with constants H i and I i ( i = 1 , ,
3) (we write H i and I i instead of ˙ H g i and ˙ I g i ) defined in (3.10)and (3.9).2 V.E. Vekslerchik Proposition 4.3.
The ‘dark-soliton’ solutions for the field equations can be presented as (cid:126)q ( (cid:126)n ) = 1det | + A ( (cid:126)n ) | (cid:18) q e (cid:104) (cid:126)ϕ , (cid:126)n (cid:105) det | + A ( (cid:126)n ) T − ν | q e (cid:104) (cid:126)ϕ , (cid:126)n (cid:105) det | + A ( (cid:126)n ) T − ν | (cid:19) and (cid:126)r ( (cid:126)n ) = 1det | + A ( (cid:126)n ) | (cid:18) r e −(cid:104) (cid:126)ϕ , (cid:126)n (cid:105) det | + A ( (cid:126)n ) T ν | r e −(cid:104) (cid:126)ϕ , (cid:126)n (cid:105) det | + A ( (cid:126)n ) T ν | (cid:19) , where q a and r a are defined in (4.19) , the vectors (cid:126)ϕ a are given by (cid:126)ϕ a = 23 (cid:88) i =1 (cid:18) ln ( g i − − ν a ) H i − ( g i +1 − ν a ) H i +1 (cid:19) (cid:126)e i , a = 1 , , while the matrix A ( (cid:126)n ) is given by A ( (cid:126)n ) = A diag (cid:0) exp (cid:104) (cid:126)φ (cid:96) , (cid:126)n (cid:105) (cid:1) N(cid:96) =1 , where (cid:126)φ (cid:96) = 23 (cid:88) i =1 (cid:32) ln g i +1 − L (cid:96) g i +1 − L − (cid:96) g i − − L − (cid:96) g i − − L (cid:96) (cid:33) (cid:126)e i and A = (cid:18) a k − L j L k (cid:19) Nj,k =1 . Here, the parameters g , g and g should be determined from (4.20) , while L k , a k ( k = 1 , . . . , N ) , ν , q , q as well as H ∗ and I ∗ in (3.10) and (3.9) are arbitrary constants. To derive the second type of soliton solutions one can use the soliton Fay identities from [20]which were obtained for the tau-functions τ = det | + AB | , σ = τ (cid:10) a | ( + BA ) − | β (cid:11) , ρ = τ (cid:10) b | ( + AB ) − | α (cid:11) , (4.21)where ( N × N )-matrices A and B are solutions of LA − AR = | α (cid:105)(cid:104) a | , RB − BL = | β (cid:105)(cid:104) b | . Here, like in the previous section, L and R are constant diagonal ( N × N )-matrices, L = diag( L , . . . , L N ) and R = diag( R , . . . , R N ), | α (cid:105) and | β (cid:105) are constant N -columns, (cid:104) a | and (cid:104) b | are N -component rows that depend on the coordinates describing the model.The shifts T ξ are defined, in this case, by T ξ (cid:104) a | = (cid:104) a | ( R − ξ ) − , T ξ (cid:104) b | = (cid:104) b | ( L − ξ )or, as a consequence, by T ξ A = A ( R − ξ ) − , T ξ B = B ( L − ξ ) . xplicit Solutions for a Nonlinear Vector Model on the Triangular Lattice 13The simplest soliton identities from [20] are( ξ − η ) τ T ξη σ = (cid:0) T ξ σ (cid:1)(cid:0) T η τ (cid:1) − (cid:0) T ξ τ (cid:1)(cid:0) T η σ (cid:1) , (4.22)( ξ − η ) ρ T ξη τ = (cid:0) T ξ τ (cid:1)(cid:0) T η ρ (cid:1) − (cid:0) T ξ ρ (cid:1)(cid:0) T η τ (cid:1) (4.23)and (cid:0) T ξ τ (cid:1)(cid:0) T η τ (cid:1) = τ T ξη τ + ρ T ξη σ. (4.24)The fact that τ , σ and ρ are solutions of (4.22)–(4.24) is enough to demonstrate that thevectors (cid:126)σ and (cid:126)ρ given by (cid:126)σ = (cid:32) E T − κ τE T κ σ (cid:33) , (cid:126)ρ = (cid:32) E − T κ τE − T − κ ρ (cid:33) , where E and E are defined by T ξ E = ˙ H ξ ˙ I ξ E , T ξ E = ( κ − ξ ) ˙ H ξ ˙ I ξ E with arbitrary κ satisfy equations (4.6) and (4.7) with A ξ,η = ( η − ξ ) ˙ H ξ ˙ H η as well as equations (4.8) and (4.9) provided ξ − κ = 1˙ H ξ ˙ I ξ . To simplify the final formulae, we introduce the diagonal matrices X ( (cid:126)n ) and Y ( (cid:126)n ), describingthe (cid:126)n -dependence of the rows (cid:104) a ( (cid:126)n ) | and (cid:104) b ( (cid:126)n ) | , (cid:104) a ( (cid:126)n ) | = (cid:104) a | X ( (cid:126)n ) , (cid:104) b ( (cid:126)n ) | = (cid:104) b | Y ( (cid:126)n ) , as well as the new rows (cid:104) ˆ a ( (cid:126)n ) | and (cid:104) ˆ b ( (cid:126)n ) | defined as (cid:104) ˆ a ( (cid:126)n ) | = (cid:104) a | (cid:2) X − ( (cid:126)n ) + BY ( (cid:126)n ) A (cid:3) − , (cid:104) ˆ b ( (cid:126)n ) | = (cid:104) b | (cid:2) Y − ( (cid:126)n ) + AX ( (cid:126)n ) B (cid:3) − . (4.25)After some simple transformations of the matrix formulae (4.21) and elimination of some‘redundant’ constants (which includes setting κ = 0) we arrive at the following result. Proposition 4.4.
The ‘bright-soliton’ solutions for the field equations can be presented as (cid:126)q ( (cid:126)n ) = (cid:32) C e (cid:104) (cid:126)ϕ , (cid:126)n (cid:105) [1 − (cid:104) ˆ a ( (cid:126)n ) | BY ( (cid:126)n ) | (cid:105) ] C e (cid:104) (cid:126)ϕ , (cid:126)n (cid:105) (cid:104) ˆ a ( (cid:126)n ) | (cid:105) (cid:33) ,(cid:126)r ( (cid:126)n ) = (cid:32) C − e −(cid:104) (cid:126)ϕ , (cid:126)n (cid:105) (cid:2) − (cid:104) ˆ b ( (cid:126)n ) | AX ( (cid:126)n ) | (cid:105) (cid:3) C − e −(cid:104) (cid:126)ϕ , (cid:126)n (cid:105) (cid:104) ˆ b ( (cid:126)n ) | (cid:105) (cid:33) , where (cid:126)ϕ = 23 (cid:88) i =1 (cid:18) ln H i +1 I i +1 H i − I i − (cid:19) (cid:126)e i , (cid:126)ϕ = 23 (cid:88) i =1 (cid:18) ln H i − H i +1 (cid:19) (cid:126)e i , the rows (cid:104) ˆ a ( (cid:126)n ) | and (cid:104) ˆ b ( (cid:126)n ) | are defined in (4.25) with X ( (cid:126)n ) = diag (cid:0) exp (cid:104) (cid:126)φ (cid:96) , (cid:126)n (cid:105) (cid:1) N(cid:96) =1 , Y ( (cid:126)n ) = diag (cid:0) exp (cid:104) (cid:126)ψ (cid:96) , (cid:126)n (cid:105) (cid:1) N(cid:96) =1 and (cid:126)φ (cid:96) = 23 (cid:88) i =1 (cid:18) ln R (cid:96) − g i − R (cid:96) − g i +1 (cid:19) (cid:126)e i , (cid:126)ψ (cid:96) = 23 (cid:88) i =1 (cid:18) ln L (cid:96) − g i +1 L (cid:96) − g i − (cid:19) (cid:126)e i , g i = 1 H i I i , | (cid:105) is the N -column with all elements equal to , (cid:104) a | and (cid:104) b | are constant N -rows, (cid:104) a | =( a , . . . , a N ) , (cid:104) b | = ( b , . . . , b N ) and constant matrices A and B are given by A = (cid:18) L j a k L j − R k (cid:19) Nj,k =1 , B = (cid:18) R j b k R j − L k (cid:19) Nj,k =1 . Here, C , C , L (cid:96) , R (cid:96) , a (cid:96) , b (cid:96) ( (cid:96) = 1 , . . . , N ) as well as H ∗ and I ∗ in (3.10) and (3.9) are arbitraryconstant parameters of the solution. In this paper we have considered the vector model on the triangular lattice. It should be notedthat we have not elaborated some special methods to take into account the vector character ofthe model or the fact that the lattice is not a rectangular one. Instead, we have used simplealgebraic calculations to demonstrate that this somewhat non-standard model can be reducedto the already known equations which are usually associated with the rectangular lattices. Ofcourse, this is a reduction. But this reduction is not a trivial one, in the sense that it leads toa rather wide range of solutions, which includes not only solutions presented above but also theso-called finite-gap quasiperiodic and many other solutions (not discussed here). Thus, one ofthe ideas behind this work is that there are much more soliton models than in the ‘standard’set of the integrable ones.However, in doing this we have met the manifestations of the peculiarities of the triangulargeometry. First of all, one should mention the rather intricate relations between the constantsand the special role of the anisotropy: note that (2.4) implies that one cannot take J = J = J or K = K = K , which can be viewed as some kind of frustration already known to occur inthe triangular lattices. Although the question of whether the restrictions (2.4), which are crucialfor our calculations, are necessary for existence of solitons is an open one, one cannot deny theimportance of the anisotropy in this model. Comparing the soliton solutions of this paper withones derived in [22] for the 3D vectors (instead of 2D) with interaction similar to (1.2), butisotropic, and without the restriction (1.3) one can conclude that it is interesting to study theinterplay between the dimensionality of the vectors, anisotropy and the length restrictions.Finally, we would like to mention the following limitations of the ansatz used in this paper.The case is that the model (1.1) with (1.2) possesses some reductions that are rather interes-ting from the viewpoint of applications. The simplest one is (cid:126)q = (cid:126)r , which after resolving therestriction (1.3), | (cid:126)q | = 1, by presenting (cid:126)q as (cid:126)q = (cos θ, sin θ ) T leads to (cid:88) i =1 J i ¯ K i sin[ θ ( (cid:126)n ) − θ ( (cid:126)n + (cid:126)e i )]1 − ¯ K i cos[ θ ( (cid:126)n ) − θ ( (cid:126)n + (cid:126)e i )] = 0 , where ¯ K i = K i / ( K i + 1 / (cid:126)q and (cid:126)r inherent in the system (2.5) and (2.6). In other words, our ansatz isxplicit Solutions for a Nonlinear Vector Model on the Triangular Lattice 15incompatible with the reduction (cid:126)r → (cid:126)q . In a similar way, our ansatz is also incompatible withanother reduction of physical interest, (cid:126)r → (cid:126)q ∗ where the star indicates the complex conjugation.Thus, a natural continuation of this work is to replace the ‘triangular’ system (3.2) with anotherone, say, a system of quad-equations having the symmetries discussed above.However, these questions are out of the scope of the present paper and may be addressed infuture studies. A Consistency of (3.2) and proof of (3.7)
Here, we derive the restrictions (3.3) and (3.4) and prove the fact that the quantity U i ( (cid:126)n ) = (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i ) , (cid:126)r ( (cid:126)n ) (cid:105) is constant or, that U i ( (cid:126)n + (cid:126)g j ) = U i ( (cid:126)n ) for all i and j ( i, j = 1 , , (cid:126)r ( (cid:126)n + g i − ) from the second equationof (3.5) multiplied by (cid:126)q ( (cid:126)n + g i +1 ) one obtains U i +1 ( (cid:126)n + (cid:126)g i − ) = U i +1 ( (cid:126)n ) . Repeating this procedure for different values of the indices, one can obtain U i ( (cid:126)n + (cid:126)g j ) = U i ( (cid:126)n ) , i (cid:54) = j. (A.1)Next, after multiplying the second equation of (3.2) by (cid:126)r ( (cid:126)n + g i − ) and (cid:126)r ( (cid:126)n + g i +1 ) one canobtain H i − (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i − ) , (cid:126)r ( (cid:126)n + (cid:126)g i +1 ) (cid:105) − H i +1 = 1 J i F i ( (cid:126)n ) U i − ( (cid:126)n ) (A.2)and H i − − H i +1 (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i +1 ) , (cid:126)r ( (cid:126)n + (cid:126)g i − ) (cid:105) = 1 J i F i ( (cid:126)n ) U i +1 ( (cid:126)n ) . (A.3)Substitution of the scalar products from (A.2) and (A.3) into the definition of F i ( (cid:126)n ), which canbe written as F i ( (cid:126)n ) = 1 K i + 2 − (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i − ) , (cid:126)r ( (cid:126)n + (cid:126)g i +1 ) (cid:105) − (cid:104) (cid:126)q ( (cid:126)n + (cid:126)g i +1 ) , (cid:126)r ( (cid:126)n + (cid:126)g i − ) (cid:105) , leads to A i ( (cid:126)n ) F i ( (cid:126)n ) = B i , where A i ( (cid:126)n ) = 1 + 1 J i U i − ( (cid:126)n ) H i − − J i U i +1 ( (cid:126)n ) H i +1 (A.4)and B i = 1 K i + 2 − H i − H i +1 − H i +1 H i − . Noting that A i ( (cid:126)n + (cid:126)g i ) = A i ( (cid:126)n ), because of (A.4) and (A.1), one can conclude that either F i ( (cid:126)n + (cid:126)g i ) = F i ( (cid:126)n ) (we do not consider this possibility as leading to rather trivial solutions) or A i ( (cid:126)n ) = B i = 0 . K i = H i − H i +1 + H i +1 H i − − , is nothing but (3.4). The first one, A i = 0, leads, after the translations (cid:126)n → (cid:126)n + (cid:126)g i ± , to U i ( (cid:126)n + (cid:126)g i ) = U i ( (cid:126)n ) which, together with (A.1), means that U i ( (cid:126)n + (cid:126)g j ) = U i ( (cid:126)n ) , i, j = 1 , , , i.e., that U i ( (cid:126)n ) are constants with respect to (cid:126)n , U i ( (cid:126)n ) = U i = const , i = 1 , , . Introducing the new constants, I i = U i /H i , one can obtain from (A.4) the constraint (3.3). Acknowledgments
We would like to thank the referees for their constructive comments and suggestions for im-provement of this paper.
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