Explicit solutions for a nonlinear vector model on the triangular lattice

We present a family of explicit solutions for a nonlinear classical vector model with anisotropic Heisenberg-like interaction on the triangular lattice.

In this paper, which is a continuation of [11,12], we study the possibilities to construct explicit solutions for nonlinear lattices with non-square geometry. Our aim is twofold. First, we want to present an application of some of the ideas and methods elaborated for models on such lattices and even on arbitrary graphs (see, for example, [1,2,6,5,4,3,8,7]). On the other hand, we want to demonstrate that some of the results obtained for the rectangular lattices can be modified to provide solutions for the models defined on the triangular ones.

The model.
We consider a nonlinear vector model defined on the triangular lattice with interaction between the nearest neighbours, S = n.n. L( n 1 , n 2 ) (2.1) described by L( n 1 , n 2 ) = J n 1 , n 2 ln 1 + K n 1 , n 2 q( n 1 ) − q( n 2 ), r( n 1 ) − r( n 2 ) (2.2) and the restriction q ( n) , r ( n) = 1 (2.3) (for all n). The vectors q and r are two-dimensional real vectors, q, r ∈ R 2 and the brackets , stand for the standard scalar product in R 2 .
It should be noted that, although we study a two-dimensional lattice, we introduce, instead of a pair of basis vectors, three coplanar vectors e 1 , e 2 and e 3 related by 3 i=1 e i = 0 and consider the lattice vectors n as a linear combinations n = 3 i=1 n i e i (see figure 1). In terms of { e i }, the set of the nearest neighbours of a lattice point n is given by { n± e i } i=1,2,3 and the discrete action (2.1) can be rewritten as J i K i q ( n + e i ) − q ( n + e i ) , r q 1 + K i q ( n) − q ( n + e i ) , r ( n) − r ( n + e i ) + q ( n − e i ) − q ( n − e i ) , r q 1 + K i q ( n) − q ( n − e i ) , r ( n) − r ( n − e i ) = 0, (2.9) 3 i=1 J i K i r ( n + e i ) − q, r ( n + e i ) r 1 + K i q ( n) − q ( n + e i ) , r ( n) − r ( n + e i ) + r ( n − e i ) − q, r ( n − e i ) r 1 + K i q ( n) − q ( n − e i ) , r ( n) − r ( n − e i ) = 0. (2.10) Namely these equations are the main subject of this work. In the following sections we present three families of explicit solutions for the system (2.9) and (2.10).

The ansatz.
The ansatz that we use in what follows is closely related to the already known ideas that may be termed 'star-triangle' (or 'star-polygon') transformation or, using the language of, e.g., [6], 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 (2.9) and (2.10) can be obtained from a more simple system of equations.

Auxiliary system.
The main steps of the proposed ansatz can be described as follows. First, we consider the original lattice { n} together with its dual, { n ± g i }, where new vectors g i are related to e i by In this equation, as well as in the rest of the paper, we use the following convention: all arithmetic operations with the indices of the vectors e i and g i and the constants J i , K i and other are understood modulo 3, In other words, we write i ± 1 bearing in mind the following: Then, we introduce a rather simple system of four-point equations relating the values of q and r at the points n, n + g i and n + g j + g k and demonstrate that each solution of the latter is at the same time a solution of the field equations (2.9) and (2.10). This system can be written as with constant H i (i = 1, 2, 3) and (note that structure of F i ( n) is that of the denominators of the summands in (2.9) and (2.10)). It turns out that the consistency of the system (3.4) implies the following restrictions on the constants J i : where I i are arbitrary constants and the following relations between K i and H i : solves equations (2.9) and (2.10) with the constants J i , K i and the vectors e i given by (3.6), (3.7) and (3.1).
The proof of this statement is rather simple except of one non-trivial fact following from the consistency of the system (3.8): the scalar products q ( n + g i ) , r ( n) do not depend on n (we prove this statement in appendix A).
After this fact is established, the rest of calculations is easy. To compute the summand in (2.9), where we express q ( n ± e i ) = q ( n ± g i+1 ∓ g i−1 ) and q ( n ± e i ) , r ( n) from the first equation from (3.8) (translated by − g i∓1 ), note that f ±i ( n) = F ( n − g i∓1 ), impose the restriction q ( n + g i ) , r ( n) = H i I i (3.12) and arrive at Now, one can easily obtain which means that the right-hand side of (2.9) vanishes automatically,

Auxiliary vectors and constants.
One of the ingredients of the proposed method to derive solutions for the field equations (2.9) and (2.10) is to use, instead the vectors e i , the vectors g i . It is natural to think of the latter as pointing to the centers of the (three of six) triangular plaquettes adjacent to the point n = 0. However, it is not necessary to endow the vectors g i with this geometrical meaning. One can consider them just as three vectors satisfying (3.1) for given set of e i . It is easy to show that equation (3.1) admits one-parametric family of solutions, with arbitrary g * . The fact that the system (3.1) does not determine g i uniquely is not important for our purposes: in the worst case we may have different solutions for different choices of g * . However, as we see in what follows, this arbitrariness does not affect the final formulae for solutions.
In a similar way, relations (3.6) and (3.6) considered as equations from which one has to find I i and H i lead to and where I * and H * are arbitrary constants or, if one uses (2.8), with arbitrary I ′ * and H ′ * . It should be noted that when we use the three vectors e i to describe the triangular lattice, instead of a two-vector basis, we bear in mind the restriction 3 i=1 e i = 0. This restriction leads to some technical problems when constructing the explicit solutions. Now, if we do not introduce the vectors g i as pointing to the centers of the plaquettes, but use (3.17) as the definition, we do not have the 'geometric' restriction:

Explicit solutions.
Here, we derive explicit solutions for the ansatz equations (3.8) and (3.9). To simplify the following formulae and to make the correspondence with the previous works we change the notation. First, we introduce the shifts T to indicate the translations: (4.1) Then, we rewrite the system of the three equations (3.8) as difference/functional equations relating q and r with T ξ q, T ξ r, T η q and T η r where ξ and η belong to the set of parameters {g 1 , g 2 , g 3 } that correspond to the translations by the vectors { g 1 , g 2 , g 3 }. This can be done by redefinition of the constants: and and can be easily bilinearized by the substitution and with arbitrary constants B ξ,η , symmetric in ξ and η (B ξ,η = B η,ξ ), which leads to the bilinear system and It should be noted that equation (4.7) turns out to be the direct consequence of the equations (4.6), the definitions (3.6) and (3.7) together with (2.3) and (3.12) which can be rewritten as These equations are, in some sense, an alternative to (4.7) and in the next sections, we will derive solutions for (4.2)-(4.3), and hence for (3.8)-(3.9), using the following Proposition 2. Any solution for the systeṁ with arbitrary skew-symmetric constants A ξ,η (A ξ,η = −A η,ξ ) provides solution for the system (4.2)-(4.3) with constants B ξ,η and C ξ,η given by (4.8) by means of (4.14) Equations (4.10) and (4.11) are known for many years. In this form, or similar ones, they have appeared in the studies of various integrable systems. Thus, one does not need to derive some special methods to solve them. In what follows we present three families of explicit solutions for (4.10)-(4.13), and, hence, for (4.2) and (4.3). In all the cases our approach is to establish the relations between equations we want to solve and the already known equations like the Jacobi identities for the Toeplitz matrices or various identities for the Cauchy-type matrices which gives us a possibility to obtain solutions by means of rather simple calculations.
The difference between these cases is different dependence of the constantsİ ξ andḢ ξ on ξ. To return from the solutions for (4.10)-(4.13) to solutions for the field equations (2.9) and (2.10) we have to invert this dependence: we have to express ξ in terms ofİ ξ andḢ ξ , i.e. to obtain the dependence of g i onİ g i andḢ g i . Then, recalling thatḢ g i = H i andİ g i = I i are functions of J i and K i given by (3.18), (3.19) and (2.7), we establish how the parameters g i depend on J i and K i .

Toeplitz solutions.
The solutions we discuss in this section are built of the determinants of the Toeplitz matrices 4.15) and the shifts defined by The action of the shifts can be 'translated' to the level of determinants ∆ M N as , (4.17) etc.
Application of the Jacobi identities to the determinants in the above formulae leads to various bilinear identities: the 'basic' one, the one-shift, the two-shift, etc. These identities are the simplest ones, from which one can deduce an infinite set of bilinear identities for the Toeplitz determinants. Using equations (4.19)-(4.23) and their direct consequences it is easy to demonstrate that functions where E is the discrete exponential function defined by which means that functions q = σ/τ and r = ρ/τ are solutions for (4.2) and (4.3). Thus, to finish the derivation of the solutions for the field equation the only thing we have to do is to rewrite the above formulae in terms of n bearing in mind that for any function f , n i f ( 0) and that in our case g i = H 2 i I i with constants I i and H i being defined in (3.18) and (3.19).
The n-dependence of the elements of the determinant (4.15) can be then presented by means of an analogue of the discrete Fourier transform as whereα ℓ and h ℓ (ℓ = 1, .., L) are arbitrary constants.
To make the following formulae more clear, we rewrite functions similar to E ( n) orα(h, n) in the exponential form using the following observation: if some function f satisfies with positive constants C i then its n-dependence can be written as Gathering the above formulae and making some trivial calculations we can formulate the main result of this section as follows.
the vector ϕ is given by and the function α m ( n) is defined as

'Dark' solitons.
Here we use some of the results of [9] where we have studied the the determinants of the so-called 'dark-soliton' matrices where L = diag (L 1 , ..., L N ) , | 1 is the N -column with all components equal to 1, a| is a Ncomponent row that depends on the coordinates describing the model, and their transformation properties with respect to the shifts defined as T ξ Ω = det |1 + T ξ A| with and the matrices T given by For our current purposes we need the two facts. First, the determinants τ satisfy the Fay-like identity (see [9]) and, secondly, the superposition of T ξ and T 1/ξ is again the shift, corresponding to zero value of the parameter, which follows from the corresponding property of the matrices T ξ , T ξ T 1/ξ = T 0 , that can be verified straightforwardly.
Using only (4.42) and (4.43), without additional referring to the structure of the matrices A, one can demonstrate that vector-functions where ν 1,2 , q 1,2 and r 1,2 are constants related by and E 1,2 are two discrete exponential functions, defined by E a (a = 1, 2), (4.46) satisfy equations (4.10) and (4.11) with as well as equations (4.12) and (4.13) provideḋ Thus, to obtain solutions for the equations we are to solve, we have just to return to the n-dependence knowing the action of T ξ . To this end, we have to take ξ ∈ {g 1 , g 2 , g 3 } where g i (parameters that, recall, correspond to the vectors g i ) should be determined from with constants H i and I i (i = 1, 2, 3) (we write H i and I i instead ofḢ g i andİ g i ) defined in (3.19) and (3.18).
Proposition 4. The 'dark-soliton' solutions for the field equations can be presented as and where q a and r a are defined in (4.45), the vectors ϕ a are given by while the matrix A ( n) is given by

'Bright' solitons.
To derive the second type of soliton solutions one can use the soliton Fay identities from [10] which were obtained for the tau-functions   Here, like in the previous section, L and R are constant diagonal N ×N -matrices, L = diag (L 1 , ..., L N ) and R = diag (R 1 , ..., R N ), | α and | β are constant N -columns, a| and b| are N -component rows that depend on the coordinates describing the model. The shifts T ξ are defined, in this case, by or, as a consequence, by The simplest soliton identities from [10] are The fact that τ , σ and ρ are solutions of (4.60)-(4.62) is enough to demonstrate that the vectors σ and ρ given by where E 1 and E 2 are defined by with arbitrary κ satisfy equations (4.10) and (4.11) with as well as equations (4.12) and (4.13) provided (4.66) To simplify the final formulae, we introduce the diagonal matrices X ( n) and Y ( n), describing the n-dependence of the rows a ( n) | and b ( n) |, as well as the new rows â ( n) | and b ( n) | defined as After some simple transformations of the matrix formulae (4.56) and elimination of some 'redundant' constants (which includes setting κ = 0) we arrive at the following result.
Proposition 5. The 'bright-soliton' solutions for the field equations can be presented as C 2 e ϕ 2 , n â ( n) |1 , (4.69)  and |1 is the N -column with all elements equal to 1, a| and b| are constant N -rows, a| = (a 1 , ..., a N ), b| = (b 1 , ..., b N ) and constant matrices A and B are given by . (4.76) .., N ) as well as H * and I * in (3.19) and (3.18) are arbitrary constant parameters of the solution.

Conclusion.
In this paper we have considered the vector model on the triangular lattice. It should be noted that we have not elaborated some special methods to take into account the vector character of the model or the fact that the lattice is not a rectangular one. Instead, we have used simple algebraic calculations to demonstrate that this somewhat non-standard model can be reduced to the already known equations which are usually associated with the rectangular lattices. Of course, this is a reduction. But this reduction is not a trivial one, in the sense that it leads to a rather wide range of solutions, which includes not only solutions presented above but also the so-called finite-gap quasiperiodic and many other solutions (not discussed here). Thus, one of the 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 triangular geometry. First of all, one should mention the rather intricate relations between the constants and the special role of the anisotropy: note that (2.8) implies that one cannot take J 1 = J 2 = J 3 or K 1 = K 2 = K 3 , which can be viewed as some kind of frustration already known to occur in the triangular lattices. Although the question of whether the restrictions (2.8), which are crucial for our calculations, are necessary for existence of solitons is an open one, one cannot deny the importance of the anisotropy in this model. Comparing the soliton solutions of this paper with ones derived in [12] for the 3D vectors (instead of 2D) with interaction similar to (2.2), but isotropic, and without the restriction (2.3) one can conclude that it is interesting to study the interplay between the dimensionality of the vectors, anisotropy and the length restrictions. However, these questions are out of the scope of the present paper and may be addressed in future studies.
Here, we derive the restrictions (3.6) and (3.7) and prove the fact that the quantity U i ( n) = q ( n + g i ) , r ( n) (A.1) is constant or, that U i ( n + g j ) = U i ( n) for all i and j (i, j = 1, 2, 3). Subtracting the first equation of (3.4) multiplied by r ( n + g i−1 ) from the second equation of (3.8) multiplied by q ( n + g i+1 ) one obtains U i+1 ( n + g i−1 ) = U i+1 ( n) . Next, after multiplying the second equation of (3.4) by r ( n + g i−1 ) and r ( n + g i+1 ) one can obtain and Substitution of the scalar products from (A.4) and (A.5) into the definition of F i ( n), which can be written as F i ( n) = 1 K i + 2 − q ( n + g i−1 ) , r ( n + g i+1 ) − q ( n + g i+1 ) , r ( n + g i−1 ) , (A.6) leads to where Noting that A i ( n + g i ) = A i ( n), because of (A.8) and (A.3), one can conclude that either F i ( n + g i ) = F i ( n) (we do not consider this possibility as leading to rather trivial solutions) or The second of these equalities, is nothing but (3.7). The first one, A i = 0, leads, after the translations n → n + g i±1 , to U i ( n + g i ) = U i ( n) which, together with (A.3), means that U i ( n + g j ) = U i ( n) (i, j = 1, 2, 3) (A. 12) i.e. that U i ( n) are constants with respect to n,