1 Introduction
###### Abstract

Generalized lattice Heisenberg magnet model is an integrable model exhibiting soliton solutions. The model is physically important for describing the magnon bound states (or soliton excitations) with arbitrary spin, in magnetic materials. In this paper, a time-discrete generalized lattice Heisenberg magnet (GLHM) model is investigated. By writing down the Lax pair representation of the time-discrete GLHM model, we present explicitly the underlying integrable structure like, the Darboux transformation and soliton solutions.

On soliton solutions of the time-discrete generalized lattice Heisenberg magnet model

H. Wajahat A. Riaz111ahmed.phyy@gmail.com and Mahmood ul Hassan 222mhassan@physics.pu.edu.pk

Department of Physics, University of Punjab, Quaid-e-Azam Campus,

Lahore-54590, Pakistan.

PACS: 02.30.Ik, 05.45.Yv
Keywords: Integrable systems, solitons, Discrete Darboux transformation

## 1 Introduction

Discrete or (lattice) integrable systems namely systems with their independent variables defined on a lattice points have received much attention by the researchers working in the field of theoretical and applied sciences. The study of discrete integrable system not only as a physical model but also in the context of numerical analysis is of particular importance in various fields ranging from pure mathematics to experimental science. Discrete integrable systems such as, Toda lattice, Volterra lattice, Ablowitz-Ladik lattice, Hirota-Miwa equation, nonlinear -model, sine-Gordon equation etc have been studied extensively in the literature [1]-[11]. Soliton solutions of these discrete systems have been computed by using various systematic methods such inverse scattering transform, Bäcklund/Darboux transformation, Hirota bilinear method etc [1]-[11], [20].

The lattice Heisenberg magnet model has been studied in many references such as [10]-[13]. It exhibits many aspects of integrability for instance Lax pair representation, higher symmetries, -matrix formulism etc. The soliton solutions have been studied by Bäcklund transformation (BT), Darboux transformation (DT), inverse scattering transform (IST) and other solution generating techniques [10]-[19].

The Lax pair representation of the time-discrete GLHM model is given by [17]

 Φmn+1 = AmnΦmn,Amn=I+λUmn, (1.1) Φm+1n = BmnΦmn,Bmn=I+hλ1−λ2Jmn+hλ21−λ2JmnUmn, (1.2)

where is an matrix and is also an eigen-function matrix. The conditions on the matrix i.e., and are assumed. The latter condition implies . The compatibility condition of the Lax pair (1.1)-(1.2) implies a zero-curvature condition i.e., which is equivalent to the equation of motion

 1h[(Am+1n)−1−(Amn)−1]+λ1−λ2(Jmn+1−Jmn)=O, (1.3)

or equivalently,

 1h(Um+1n−Umn)=Jmn+1−Jmn (1.4)

The relation is satisfied if we choose . Substituting this expression into equation (1.4), we obtain

 1h(Um+1n−Umn)=Δ+n[2˙ıamnUm+1n−1(Umn+Um+1n−1)−1+2bmn(Umn+Um+1n−1)−1], (1.5)

where . For , we get a simplest case and express the equation (1.5) as

 1h(Um+1n−Umn)=Δ+n⎡⎣aUmn×Um+1n−11+Umn.Um+1n−1+bmnUmn+Um+1n−11+Umn.Um+1n−1⎤⎦,(Umn)2=1. (1.6)

It should be noted that the generalization of the lattice Heisenberg model (1.4), (1.5) was studied by Tsuchida [17]. In the case of matrices of size , it reduces to the well-known vector lattice Heisenberg chain (1.6). The Bäcklund transformations for the latter are sufficiently well studied, in particular, their derivation is given in [17], with references to earlier works. However, the problem of describing Bäcklund/Darboux transformations for the general matrix case (1.4), (1.5) is left open [17]. It is this problem that is considered in the present work.

In this paper, we present a systematic approach to find the soliton solutions of the time-discrete GLHM model (1.4). We define Darboux transformation (DT) on the solution to the Lax pair and the solutions of the matrix generalization of the time-discrete GLHM model defined by equation (1.4) with respect to the matrices , or, after reduction, by one equation (1.5) with respect to the matrix .

Darboux transformation is one of the powerful and effective technique used to compute solutions of a given nonlinear integrable equation in soliton theory. The main idea of this method is that, a new solution to the Lax pair (i.e., pair of linear equations associated with nonlinear integrable equation) can be obtain from the old solution by means of Darboux matrix. The covariance of the Lax pair under the Darboux transformation requires that the new solution satisfies the same Lax pair such that the relationship between new and old solutions to the Lax pair and the solutions to the nonlinear integrable equation can be built. Hence, one can find the soliton solutions to the nonlinear integrable equation by solving a Lax pair with the given seed (or trivial) solutions. Various integrable equations have been studied successfully by means of Darboux transformation and obtained the soliton solutions have been computed. The solutions are expressed in terms of Wronskian, quasi-Grammian and quasi-determinants in the literature [8]-[9], [20]-[24].

This paper is organized as follows. Section 2, contains the derivation of the DT for the matrix generalization of the time-discrete GLHM model (1.4), (1.5). Furthermore, the solutions obtained by DT are expressed in qusideterminant form. In section 3, soliton solutions for the general and simplest case of the time-discrete GLHM model are obtained. Section 4, is devoted for concluding remarks.

## 2 Discrete Darboux transformation

In what follows, we apply DT on the Lax pair (1.1)-(1.2) of the time-discrete GLHM model (1.4) to obtain soliton solutions. We define a DT on the solutions to the Lax pair equations (1.1)-(1.2) by means of a discrete Darboux matrix . The discrete Darboux matrix acts on the solution of the Lax pair (1.1)-(1.2) to give another solution i.e.,

 Φmn[1]=DmnΦmn. (2.1)

The covariance of the Lax pair (1.1)-(1.2) under the DT requires that the new solution satisfies the same Lax pair equations with the new matrices i.e.,

 Amn[1] = I+λUmn[1], (2.2) Bmn[1] = I+hλ1−λ2Jmn[1]+hλ21−λ2Jmn[1]Umn[1], (2.3)

By using (1.1)-(1.2), equations (2.2)-(2.3) imply that the discrete Darboux matrix satisfies the following discrete Darboux-Lax equations as

 Amn[1]Dmn=Dmn+1Amn,Bmn[1]Dmn=Dm+1nBmn (2.4)

We are interested in finding a DT on the matrices . For this, we make the ansatz for the Darboux matrix such as where is the auxiliary matrix and is the identity matrix. By substituting the latter expression of in equation (2.4), the coefficients of yields the DT on the matrices , as

 Umn[1] = Umn−(Qmn+1−Qmn), (2.5) Jmn[1] = Jmn−1h(Qm+1n−Qmn), (2.6)

with the following conditions on the Darboux matrix

 (Qmn+1−Qmn)Qmn = UmnQmn−Qmn+1Umn, (2.7) 1h(Qm+1n−Qmn)(I−(Qmn)2) = [Qmn,Jmn(Qmn+Umn)]+. (2.8)

where . In what follows, we show that for , where is the particular matrix solution to the Lax pair (1.1)-(1.2) which is constructed by wave function for different values of , whereas the matrix is an diagonal matrix with distinct eigenvalues (), . Take constant column basis vectors , so that the invertible matrix can be defined as , such that each in the matrix is a column solution to the Lax pair (1.1)-(1.2). For , we have

 |θi⟩mn+1 = |θi⟩mn+λiUmn|θi⟩mn, (2.9) |θi⟩m+1n = |θi⟩mn+hλi1−λ2iJmn|θi⟩mn+hλ2i1−λ2iJmnUmn|θi⟩mn, (2.10)

For , the Lax pair (2.9)-(2.10) reduce to the generalized matrix form as

 Θmn+1 = Θmn+UmnΘmnΛ, (2.11) Θm+1n = Θmn+hJmnΘmnΛ(I−Λ2)−1+hJmnUmnΘmnΛ2(I−Λ2)−1, (2.12)

where is a particular matrix solution to the Lax pair (1.1)-(1.2). Let us define a matrix in terms of an invertible matrix , i.e. . Now, we show that the latter expression of the matrix satisfies the set of equations (2.7)-(2.8). To do this, let us check the first condition as

 (Qmn+1−Qmn)Qmn, =(Θmn+1Λ−1(Θmn+1)−1−ΘmnΛ−1(Θmn)−1)ΘmnΛ−1(Θmn)−1, =UmnQmn−Qmn+1Umn, (2.13)

which is equation (2.7). Similarly for the second condition, we have

 1h(Qm+1n−Qmn)(I−(Qmn)2) =1h(Θm+1nΛ−1(Θm+1n)−1−ΘmnΛ−1(Θmn)−1+Θm+1nΛ−1(Θmn)−1−Θm+1nΛ−1(Θmn)−1) ×Θmn(I−Λ−2)(Θmn)−1, =1h(Θm+1nΛ−1(Θm+1n)−1(Θmn−Θm+1n)−(Θmn−Θm+1n)Λ−1)(I−Λ−2)(Θmn)−1, =−Θm+1nΛ−1(Θm+1n)−1(JmnΘmnΛ(I−Λ2)−1+JmnUmnΘmnΛ2(I−Λ2)−1)(I−Λ−2)(Θmn)−1 +(JmnΘmnΛ(I−Λ2)−1+JmnUmnΘmnΛ2(I−Λ2)−1)Λ−1(I−Λ−2)(Θmn)−1, =[Qmn,Jmn(Qmn+Smn)]+. (2.14)

which is equation (2.8). Therefore, we have established that the DT on the matrix solutions is given by

 Φmn[1] = (λ−1I−ΘmnΛ−1(Θmn)−1)Φmn, (2.15) Umn[1] = Umn−(Θmn+1Λ−1(Θmn+1)−1−ΘmnΛ−1(Θmn)−1), (2.16) = Θmn+1Λ−1(Θmn+1)−1UmnΘmnΛ−1(Θmn)−1.

At this stage, we can say that DT (2.15)-(2.16) preserves the system i.e., if are the solutions to the Lax pair (1.1)-(1.2) and GLHM model (1.4) respectively, then (that correspond to multi-soliton solutions) are also the solutions of the same equations. The DT (2.16) is also consistent with the reduction (1.5). For , it seems appropriate here to express the solutions in terms of quasideterminants.333In this paper, we will use quasideterminants that are expanded about matrix. The quasideterminant expression of expanded about matrix is given as (2.17) For further details see [22]-[25]. The matrix solution to the Lax pair (1.1)-(1.2) with the particular matrix solution in terms of quasideterminant can be expressed as

 Ψmn[1] ≡ DmnΨmn=(λ−1I−ΘmnΛ−1(Θmn)−1)Φmn, (2.18) = ∣∣∣ΘmnΦmnΘmnΛ−1\framebox$λ−1Φmn$∣∣∣.

And the one-fold Darboux transformation on the matrix field of the time-discrete GLHM is

 Umn[1] = Qmn+1Umn(Qmn)−1=Θmn+1Λ−1(Θmn+1)−1Umn(ΘmnΛ−1(Θmn)−1)−1, (2.19) = ∣∣∣Θmn+1IΘmn+1Λ−1\framebox$O$∣∣∣Umn∣∣∣ΘmnIΘmnΛ−1\framebox$O$∣∣∣−1.

where is the null matrix and is the identity matrix. The results obtained in (2.18)-(2.19) can be extended and generalized to -fold DT. For the matrix solutions at to the Lax pair (1.1)-(1.2), the -times repeated DT in terms of quasideterminant is written as

 Φmn[K] = K∏k=1(λI−Qmn[K−k])Φmn, (2.20) = K∏k=1(λI−Θmn[K−k]Λ−1K−k(Θmn[K−k])−1)Φmn, =

Similarly the quasideterminant expression for is

 Umn[K] = ∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣Θmn+1,1Θmn+1,2⋯Θmn+1,KIΘmn+1,1Λ−11Θmn+1,2Λ−12⋯Θmn+1,KΛ−1KO⋮⋮⋱⋮⋮Θmn+1,1Λ−K+11Θmn+1,2Λ−K+12⋯Θmn+1,KΛ−K+1KOΘmn+1,1Λ−K1Θmn+1,2Λ−K2⋯Θmn+1,KΛ−KK\framebox$O$∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣ (2.21) × Umn∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣Θmn,1Θmn,2⋯Θmn,KIΘmn,1Λ−11Θmn,2Λ−12⋯Θmn,KΛ−1KO⋮⋮⋱⋮⋮Θmn,1Λ−K+11Θmn,2Λ−K+12⋯Θmn,KΛ−K+1KOΘmn,1Λ−K1Θmn,2Λ−K2⋯Θmn,KΛ−KK\framebox$O$∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣−1.

Equations (2.20) and (2.21) represent respectively, the required th quasideterminant solutions to the Lax pair and of the time-discrete GLHM model. These results can be easily proved by induction.

## 3 Soliton solutions

In this section, we obtain the soliton solutions from a seed (trivial) solution by solving the Lax pair of the time-discrete GLHM model. For this, we re-write the matrix from (2.21) in a more convenient form as follows

 Qmn(K)=∣∣ ∣∣GmnI(K)˜Gmn\framebox$O$∣∣ ∣∣, (3.1)

where and are and matrices respectively. These matrices are given by

 I(K) = (IO⋯O)T, ˜Gmn = (Θmn,1Λ−K1Θmn,2Λ−K2⋯Θmn,KΛ−KK), Gmn = ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Θmn,1Θmn,2⋯Θmn,KΘmn,1Λ−11Θmn,2Λ−12⋯Θmn,KΛ−1K⋮⋮⋱⋮Θmn,1Λ−K+11Θmn,2Λ−K+12⋯Θmn,KΛ−K+1K⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (3.2)

And the components of the matrix can be decomposed as

 Qmn,ij(K) = (∣∣ ∣∣GmnI(K)˜Gmn\framebox$O$∣∣ ∣∣)ij=∣∣ ∣∣GmnI(K)j(˜Gmn)i\framebox$0$∣∣ ∣∣, (3.3) = −det(Gmn)ijdet(Gmn),i,j=1,2,...,K.

where represent -th row and -th column of the matrices respectively. For the simplest matrix of size , the matrix can be expressed as

 Qmn(K)≡⎛⎝Qmn,11(K)Qmn,12(K)Qmn,21(K)Qmn,22(K)⎞⎠=∣∣ ∣∣GmnI(K)˜Gmn\framebox$O2$∣∣ ∣∣, (3.4)

with the elements given by

 Qmn,ij(K) = ∣∣ ∣∣GmnI(K)j(˜Gmn)i\framebox$0$∣∣ ∣∣=−det(Gmn)ijdet(Gmn),i,j=1,2. (3.5)

For one soliton , we have

 I(1) = ˜Gmn = Θmn,1Λ−11=⎛⎝λ−11θmn,11(1)¯λ−11θmn,12(2)λ−11θmn,21(1)¯λ−11θmn,22(2)⎞⎠. (3.6)

By using equation (3) in (3.5), the matrix element of the matrix can be computed as

 Qmn,12(1) = ∣∣ ∣∣GmnI(1)2(˜Gmn)1\framebox$O2$∣∣ ∣∣=∣∣ ∣ ∣ ∣∣θmn,11(1)θmn,12(2)0θmn,21(1)θmn,22(2)1λ−11θmn,11(1)¯λ−11θmn,12(2)\framebox0∣∣ ∣ ∣ ∣∣, = −det⎛⎝θmn,11(1)θmn,12(2)λ−11θmn,11(1)¯λ−11θmn,12(2)⎞⎠det⎛⎝θmn,11(1)θmn,12(2)θmn,21(1)θmn,22(2)⎞⎠=(λ−11−¯λ−11)θmn,11(1)θmn,12(2)θmn,11(1)θmn,22(2)−θmn,12(2)θmn,21(1).

Likewise,

 Qmn,21(1)=−(λ−11−¯λ−11)θmn,21(1)θmn,22(2)θmn,11(1)θmn,22(2)−θmn,12(2)θmn,21(1). (3.8)

Similarly, we have

 Qmn,11(1) = −λ−11θmn,11(1)θmn,22(2)−¯λ−11θmn,12(2)θmn,21(1)θmn,11(1)θmn,22(2)−θmn,12(2)θmn,21(1), Q(1)n,22 = −¯λ−11θmn,11(1)θmn,22(2)−λ−11θmn,12(2)θmn,21(1)θmn,11(1)θmn,22(2)−θmn,12(2)θmn,21(1). (3.9)

To obtain an explicit form of the soliton solution for the general case, let us take as a seed solution, where are real constants and , so that solution to the Lax pair (1.1)-(1.2) is given by

 Φmn=(ZmnpOOZmnN−p), (3.10)

where are and matrices respectively, whereas and appearing in the latter expressions, denote the discrete indices. And

 ζi(λ)=(1+˙ıciλ)n(1−˙ıhc−1iλ1−λ2+hλ21−λ2)m. (3.11)

For the matrix of size , take as the seed solution, so that a trivial calculation yields a matrix solution of the Lax pair (1.1)-(1.2), given by

 Φmn=(ζ(λ)¯ζ(λ)),ζ(λ)=(1+˙ıcλ)n(1−˙