During the past few decades, there has been a lot of interest in the study of continuous and lattice Heisenberg magnet models. The continuous Heisenberg magnet model is completely integrable and exhibits the exact soliton solutions. Similarly, the lattice Heisenberg magnet model also preserves the integrability. The soliton solutions of this model have been studied using the inverse scattering transform, Bäcklund transformation, Darboux transformation and other solution-generating methods (see, e.g. [1–9]). The lattice Heisenberg magnet model has been studied in many works (see [10–13]). The existence of a Lax pair, Bäcklund transformation and other symmetries of the lattice Heisenberg magnet model explains many aspects of integrability [10–17]. Darboux transformation of the generalized lattice Heisenberg magnet model is studied in [25] and soliton solutions are presented.

Discrete integrable systems have received much attention from modern researchers. Many techniques, such as the Darboux transformation, the Hirota method, the Bäcklund transformation, etc, have been employed to calculate the exact solutions of many nonlinear partial differential equations ([30–36]). Binary Darboux transformation is a well-known technique used to compute the Grammian-type multisolitons of integrable systems [19, 33]. The general mechanism of this method is to keep both the spectral problem and the corresponding adjoint spectral problem associated with the nonlinear equations, which are invariant with respect to the action of the binary Darboux transformation. Furthermore, the solutions can be expressed in terms of Grammian, quasi-Grammian and quasideterminants in the literature [18, 20, 33, 37, 38].

In this paper, we study the discrete Darboux and binary Darboux transformation of the generalized lattice Heisenberg magnet (GLHM) model. For this purpose, we operate the discrete Darboux matrix on a Lax pair of GLHM models for both the direct and adjoint space to calculate the multi-soliton solutions. For the representation of solutions, we use the quasideterminant approach. Furthermore, by the iteration of binary Darboux transformation, we derive the general expressions of multi quasi-Grammian solutions. Finally, we obtain the explicit solutions for the GLHM model based upon Lie group SU(2) transformation and present the solutions, which also include the soliton solution.

The Lax pair of the GLHM model is given as,having matrices A_n and B_n given by,where the matrix function U_n ≡ U_n(t) take values from Lie group ${ \mathcal G }$ in the Lie algebra g and $Psi$_n ≡ $Psi$_n(λ) is an N × N eigen matrix, which depends upon variable n written in subscripts defined over a lattice. The matrix function U_n is subjected to constraints given by ${U}_{n}^{2}=I$ and J_nU_n = U_n−1J_n, which also implies J_nA_n = A_n−1J_n. The compatibility condition dA_n/dt + A_nB_n − B_n+1A_n = 0, is operated on (2.1), which gives the equation of motion of the GLHM model given by,Equation of motion (2.3) implies that,which is satisfied if we take,By substituting equation (2.5) in (2.3), we derive the following form of the equation of motion:where Δ_nf_n = f_n+1 − f_n. For N = 2, we have the simplest 2 × 2 case of Lie group SU(2), for which the matrix U_n is expressed as ${U}_{n}={U}_{n}^{a}{\sigma }_{a}$, where σ_a are the familiar Pauli matrices and the constraint on the matrix U_n becomes ${U}_{n}^{2}=I$. I is the 2 × 2 identity matrix. We substitute $2{\left({U}_{n}+{U}_{n-1}\right)}^{-1}=({U}_{n}+{U}_{n-1})/(1+{U}_{n}{U}_{n-1})$ in (2.6) and are able to express the equation of motion in vector notation as,

Darboux transformation is an important tool to find solutions of integrable systems represented by differential equations, partial differential equations and differential-difference equations (for details see [20–29]). We then define the Darboux transformation on the Lax pair (2.1) by using the N × N Darboux matrix D_n(λ) to calculate the soliton solutions. The Darboux matrix transforms the matrix solution from the space W to new space $\widetilde{W}$, i.e.The one-fold Darboux transformation on matrix solution $Psi$_n is defined as,where D_n(λ) is the Darboux matrix. The new transformed solution $Psi$_n[1] satisfies the following Lax pair (2.1) as,having A_n[1] and B_n[1] as,where I is the identity matrix. In order to obtain the Darboux transformation on matrix solution U_n[1], we define the Darboux matrix as,where I is the N × N identity matrix and Q_n is the auxiliary matrix of N × N order, which is yet to be found. The choice for Q_n is ${Q}_{n}={H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}$, where H_n is the distinct matrix solution of the Lax pair (2.1) having order N × N, which can be obtained by using i-eigenvector functions ${\rm{\Psi }}({\lambda }_{{\rm{i}}}){\left|\sigma \right\rangle }_{i}$ evaluated at λ_i, i = 1,…,N, whereas matrix Λ is a diagonal matrix of order N × N having eigenvalues λ₁, λ₂,…,λ_N. Therefore, matrix H_n can be defined as,,evaluated at,Using (3.6) and (3.7), the Lax pair (2.1) can be written in matrix form as, Based upon the above results, we can prove the following theorems.

Theorem 1.Under the action of Darboux transformation (3.5), the new solution (3.4) has the identical form as U_n in equation (2.2), provided that matrix Q_n fulfills the following conditions:

(3.10)$\begin{eqnarray}{U}_{n}[1]={U}_{n}-\left({Q}_{n+1}-{Q}_{n}\right),\end{eqnarray}$

(3.11)$\begin{eqnarray}\left({Q}_{n+1}-{Q}_{n}\right){Q}_{n}={U}_{n}{Q}_{n}-{Q}_{n+1}{U}_{n}.\end{eqnarray}$

Proof.The relation between the Darboux transformed solution ${U}_{n}[1]$ and the untransformed solution U_n is developed and defined in equation (3.10). We then have to show that the choice of matrix ${Q}_{n}={H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}$ satisfies the condition (3.11), i.e.

$\begin{eqnarray*}\begin{array}{l}\left({Q}_{n+1}-{Q}_{n}\right){Q}_{n}\\ \quad =\left({H}_{n+1}{{\rm{\Lambda }}}^{-1}{H}_{n+1}^{-1}-{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}\right){H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1},\\ \quad ={H}_{n+1}{{\rm{\Lambda }}}^{-1}{H}_{n+1}^{-1}{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}-{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}\\ \quad +{H}_{n+1}{{\rm{\Lambda }}}^{-2}{H}_{n}^{-1}-{H}_{n+1}{{\rm{\Lambda }}}^{-2}{H}_{n}^{-1},\\ \quad =\left({H}_{n+1}-{H}_{n}\right){{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}\\ \quad -{H}_{n+1}{{\rm{\Lambda }}}^{-1}{H}_{n+1}^{-1}\left({H}_{n+1}-{H}_{n}\right){{\rm{\Lambda }}}^{-1}{H}_{n}^{-1},\\ \quad ={U}_{n}{Q}_{n}-{Q}_{n+1}{U}_{n}.,\end{array}\end{eqnarray*}$

which is equivalent to (3.11). Therefore, the proof is complete.

Theorem 2.Under the action of Darboux transformation (3.5), the new solution (3.4 ) has the identical form as J_n in equation (2.2 ), provided that matrix Q_n fulfills the following conditions:

(3.12)$\begin{eqnarray}{J}_{n}[1]={J}_{n}-\displaystyle \frac{{\rm{d}}{Q}_{n}}{{\rm{d}}{t}},\end{eqnarray}$

(3.13)$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{Q}_{n}}{{\rm{d}}{t}}\left(I-{Q}_{n}^{2}\right)=\left[{Q}_{n},{J}_{n}\left({Q}_{n}+{U}_{n}\right)\right].\end{eqnarray}$

Proof.The relation between the Darboux transformed solution ${J}_{n}[1]$ and the untransformed solution J_n is developed and defined in equation (3.12). We then have to show that the choice of matrix ${Q}_{n}={H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}$ satisfies the condition (3.13). For this, we operate $\tfrac{{\rm{d}}}{{\rm{d}}{t}}$ on matrix $(I-{Q}_{n}^{2})$ as,

$\begin{eqnarray*}\begin{array}{l}\displaystyle \frac{{\rm{d}}{Q}_{n}}{{\rm{d}}{t}}\left(I-{Q}_{n}^{2}\right)\\ \quad =\left(\displaystyle \frac{{\rm{d}}}{{\rm{d}}{t}}{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}\right)\left(I-{H}_{n}{{\rm{\Lambda }}}^{-2}{H}_{n}^{-1}\right),\\ \quad =\displaystyle \frac{{\rm{d}}{H}_{n}}{{\rm{d}}{t}}{{\rm{\Lambda }}}^{-1}\left(I-{{\rm{\Lambda }}}^{-2}\right){H}_{n}^{-1}\\ \quad -{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}\displaystyle \frac{{\rm{d}}{H}_{n}}{{\rm{d}}{t}}\left(I-{{\rm{\Lambda }}}^{-2}\right){H}_{n}^{-1},\\ \quad =\left[{J}_{n}{H}_{n}{{\rm{\Lambda }}}^{-1}{\left(I-{{\rm{\Lambda }}}^{-2}\right)}^{-1}+{J}_{n}{U}_{n}{H}_{n}{{\rm{\Lambda }}}^{2}{\left(I-{{\rm{\Lambda }}}^{2}\right)}^{-1}\right]\\ \quad {{\rm{\Lambda }}}^{-1}\left(I-{{\rm{\Lambda }}}^{-2}\right){H}_{n}^{-1}-\\ {H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}\left[{J}_{n}{H}_{n}{\rm{\Lambda }}{\left(I-{{\rm{\Lambda }}}^{2}\right)}^{-1}\right.\\ \quad \left.+{J}_{n}{U}_{n}{H}_{n}{{\rm{\Lambda }}}^{2}{\left(I-{{\rm{\Lambda }}}^{2}\right)}^{-1}\right]\left(I-{{\rm{\Lambda }}}^{-2}\right){H}_{n}^{-1},\\ \quad =\left[{Q}_{n},{J}_{n}\left({Q}_{n}+{U}_{n}\right)\right],\end{array}\end{eqnarray*}$

which is equivalent to (3.13). Therefore, the proof is complete.

Remark 1.Thus, the matrix ${Q}_{n}={H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}$ is a good choice, which satisfies the conditions imposed by the Darboux transformation. Thus, the Darboux transformation preserves the system, i.e. if ${{\rm{\Psi }}}_{n},{U}_{n}$ and J_n, respectively, are the solutions of the linear system (2.1) and (2.2) and the equation of motion (2.4), then ${{\rm{\Psi }}}_{n}[1],{U}_{n}[1]$ and ${W}_{n}[1]$ are also the solutions of the same equations.

In order to study the solutions we use the technique known as quasideterminants given by,For details see [39, 40]. Thus, we can write the Darboux transformation on matrix solution $Psi$_n as,For the next iteration of Darboux transformation, take Q_n,1 and Q_n,2 as the two particular solutions of the Lax pair (3.3) and (3.4) at ${\rm{\Lambda }}={{\rm{\Lambda }}}_{1}^{-1}$ and ${\rm{\Lambda }}={{\rm{\Lambda }}}_{2}^{-1}$, respectively. The two-fold Darboux transformation on $Psi$_n[1] is defined as,where ${Q}_{n}[1]={H}_{n,1}{{\rm{\Lambda }}}_{1}^{-1}{H}_{n,1}^{-1}$, ${Q}_{n}[2]={H}_{n}[2]{{\rm{\Lambda }}}_{2}^{-1}{\left({H}_{n}[2]\right)}^{-1}$. Also, H_n[2] is written as,Using (3.14) and (3.16) in (3.15), we obtain the following:where we have used a homological relation in the second step and a noncommutative Jacobi identity in the last step.¹(¹For a general quasideterminant expanded about N × N matrix D, we have $\left|\begin{array}{c}E\qquad F\qquad G\\ H\qquad A\qquad B\\ J\qquad C\qquad \boxed{D}\end{array}\right|=\left|\begin{array}{c}E\qquad G\\ J\qquad \boxed{D}\end{array}\right|-\left|\begin{array}{c}E\qquad F\\ J\qquad \boxed{C}\end{array}\right|{\left|\begin{array}{c}E\qquad F\\ H\qquad \boxed{A}\end{array}\right|}^{-1}\left|\begin{array}{c}E\qquad G\\ H\qquad \boxed{B}\end{array}\right|.$ From the noncommutative Jacobi identity, we obtain the homological relation $\left|\begin{array}{c}E\qquad F\qquad G\\ H\qquad A\qquad \boxed{B}\\ J\qquad C\qquad D\end{array}\right|=\left|\begin{array}{c}E\qquad F\qquad O\\ H\qquad A\qquad \boxed{O}\\ J\qquad C\qquad I\end{array}\right|\left|\begin{array}{c}E\qquad F\qquad G\\ H\qquad A\qquad B\\ J\qquad C\qquad \boxed{D}\end{array}\right|,$ where O and I denote the null and identity matrices, respectively.) Similarly, the K-fold Darboux transformation is given by,

The expression (3.10) can then be expressed as,The result can be generalized to K-times Darboux transformation as,The expressions given by equations (3.17) and (3.19) are the Kth solutions of the GLHM model and these results can easily be derived through induction. We then construct the adjoint Darboux transformation. The adjoint Lax pair is obtained by taking the formal adjoint of the linear equations (2.1) written as,with ${A}_{n}^{\dagger }$ and ${B}_{n}^{\dagger }$ given by,where η is a spectral parameter and Φ_n is an invertible N × N matrix field in the adjoint space W^†. The Darboux matrix D_n(η) transforms the matrix solution Φ_n in space W^† to a new matrix solution ${\tilde{{\rm{\Phi }}}}_{n}$ in $\widetilde{{W}^{\dagger }}$ i.e.Based upon the above facts, we can write Darboux transformation Φ_n as,where S_n is the N × N matrix that is to be determined and I is N × N identity matrix. The covariance of the Lax pair under the Darboux transformation requires that the new solution ${\widetilde{{\rm{\Phi }}}}_{n}$ satisfies the Lax pair (3.20) given by,By operating the Darboux transformation (3.23) on (3.24), we obtain the Darboux transformed matrix functions ${U}_{n}^{\dagger }$ and ${J}_{n}^{\dagger }$ as,The matrix S_n can be constructed from the eigen matrices of the Lax pair and we take S_n to be ${S}_{n}={M}_{n}{{\rm{\Xi }}}^{-1}{M}_{n}^{-1}$, where Ξ= diag(η₁,…,η_n) is the eigenvalue matrix. The particular matrix solution M_n of the Lax pair (3.20) is an invertible N × N matrix, which is given by,Each column ${\left|{{\rm{\Phi }}}_{{\rm{i}}}\right\rangle }_{n}={{\rm{\Phi }}}_{n}({\eta }_{{\rm{i}}})\left|{e}_{i}\right\rangle $ in M_n is a column solution of the Lax pair (3.20). The K-fold Darboux transformation on the matrix solution and matrix function Φ_n, ${U}_{n}^{\dagger }$ can be written as,

Similarly, the quasideterminant of ${U}_{n}^{\dagger }[K]$ is,Equations (3.27) and (3.28) are the Kth quasideterminant solutions of the GLHM model for the adjoint space.

In order to define the binary Darboux transformation, we consider the hat space $\hat{W}$, which is a copied version of direct space W, so the corresponding solutions are ${\hat{{\rm{\Phi }}}}_{n}$ ∈$\hat{W}$. Therefore, the equation of motion and the compatibility condition will have the identical form as that for the direct space given by,having ${\hat{A}}_{n}$ and ${\hat{B}}_{n}$ are,The specific solutions for the direct and adjoint spaces are H_n and S_n, respectively. Thus, the corresponding solutions for $\hat{W}$ are ${\hat{H}}_{n}$ ∈$\hat{W}$ and ${\hat{{\rm{\Phi }}}}_{n}$ ∈ ${\hat{W}}_{n}^{\dagger }.$ Also assuming that ${\rm{i}}({\hat{H}}_{n})$ ∈ ${\widetilde{W}}^{\dagger },$ we can then write the transformation as,Since ${{\rm{\Phi }}}_{n}\in {W}_{n}^{\dagger }$, we have,Also from ${D}_{n}^{\dagger }(\lambda )({\rm{i}}({H}_{n}))=0$, we obtain ${\rm{i}}({H}_{n})={M}_{n}^{(-1)\dagger }$ and similarly ${\rm{i}}({\hat{M}}_{n})={\hat{M}}_{n}^{(-1)\dagger }$. Therefore, from the above equations we can write,andwhere ${D}_{n}(\lambda )={\lambda }^{-1}I-{H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}$. By substituting the expression of D_n(λ) in equation (4.5), we obtain,where the algebraic potential Δ_n is defined as,Similarly, for the adjoint space matrix ${\hat{M}}_{n}$ is written as,where,By writing equations (4.7) and (4.9) in matrix form for the solutions H_n and M_n, we obtain the condition on Δ_n which is given by,where the Δ_n matrix is given by,Therefore, the required potential is expressed in terms of particular matrix solutions to the Lax pair as well as to the adjoint Lax pair of the GLHM model. We then define the Darboux matrix in hat space:where,We may write above formalism as the following Darboux maps:When we apply ${\hat{D}}_{n}(\lambda )$ on equation (4.1), the equation must be covariant, i.e.Matrix ${\hat{Q}}_{n}$ is defined as,The Darboux transformation on matrix field ${\hat{{\rm{\Psi }}}}_{n}$ and ${\hat{U}}_{n}$ in hat space ${\hat{W}}_{n}$ is,We then define the standard binary Darboux transformation from equation (4.14), which relates ${\hat{{\rm{\Psi }}}}_{n}$ and $Psi$_n as,which implies that,We then operate the standard binary Darboux transformation on matrix solution $Psi$_n as,By using the expression (4.6) in the above equation, we obtain,By substituting the expression of Δ_n(H_n, Φ_n)Λ⁻¹ from equation (4.10) in the equation (4.19), we obtain,By using equation (4.9), the above expression becomes,which is the standard form of binary Darboux transformation on matrix $Psi$_n. In terms of quasideterminant, the above expression can be expressed as,This is known as the quasi-Grammian solution of the system. Similarly, for the adjoint space $\hat{{\rm{\Phi }}}$ ∈${\hat{W}}^{\dagger }$ we obtain,The standard binary Darboux transformation on the solution of GLHM model U_n, is given by,where ${\hat{Q}}_{n+1}^{-1}={\hat{H}}_{n+1}{{\rm{\Xi }}}^{\dagger }{\hat{H}}_{n+1}^{-1}$, ${\hat{Q}}_{n}={\hat{H}}_{n}{{\rm{\Xi }}}^{(-1)\dagger }{\hat{H}}_{n}^{-1}$, ${Q}_{n+1}={H}_{n+1}{{\rm{\Lambda }}}^{-1}{H}_{n+1}^{-1}$, ${Q}_{n}^{-1}={H}_{n}{\rm{\Lambda }}{H}_{n}^{-1}$, and so equation (4.23) becomes,This is in fact a product of quasideterminants, i.e.This expression can be further simplified when we introduce potential Δ_n instead of matrices in the hat space. By using equation (4.6), the above equation becomes,By substituting equation (4.10) in the above expression, we obtain,In terms of quasideterminants, the above expression can be written as,Similarly, we can calculate the Kth iteration of $Psi$_n through the iteration of binary Darboux transformation given by,The Kth iteration for the adjoint binary Darboux transformation is given by,Similarly, ${\hat{U}}_{n}[K]$ can be written as,

Similarly, using the iteration process we can calculate the quasideterminant solutions for ${U}_{n}^{\dagger }$.

Remark 2.Therefore, we can calculate the Grammian-type solutions for the GLHM model by using standard binary Darboux transformation. In addition, the potential can be expressed in the form of quasideterminants. Thus, by developing the binary Darboux transformation in terms of spectral parameters, we can obtain expressions of matrix solutions in the form of Grammian-type quasideterminants that have a different form as calculated using elementary Darboux transformation.

In this section, we consider the GLHM model based on the Lie group SU(2) and obtain the soliton solutions by using the binary Darboux transformation. To obtain an explicit expression for the soliton solution in the general N × N case, we take the seed solution,where c_i are real constants and Tr U_n = 0. A trivial calculation then yields a matrix solution $Psi$_n of the Lax pair (2.1) with the form,where,are respectively p × p and (N − p) × (N − p) matrices. Here, n in the subscript is a discrete index. We then take the seed solution for the case N = 2, which is given as,Thus, the solution of the linear system (2.1) can be expressed as,where,The particular matrix solution H_n of the direct Lax pair can be written by using the above equation (5.2) as,The expression for ${Q}_{n}={H}_{n}{{\rm{\Lambda }}}^{-1}{H}_{n}^{-1}$ by using ${\rm{\Lambda }}=\left(\begin{array}{cc}\lambda & 0\\ 0 & \bar{\lambda }\end{array}\right)$ becomes,where,Similarly, the particular matrix solution M_n for adjoint space can be written as,where,In order to obtain the expression for ${\hat{U}}_{n}$, we start from the definition of Δ_n(H_n, M_n) given in (4.11) and by using (5.4), (5.7), we obtain,where,Then, we take, where,The matrix field in hat space is given by,where,The expressions (5.11) are presented in figures 1 and 2.

In this paper, we have composed the discrete Darboux transformation of the GLHM model not only for the direct space, but also for the adjoint space, and also calculated the standard binary Darboux transformation of the model. By iterating the standard binary Darboux transformation we have obtained the multi-Grammian solutions in terms of quasideterminants. We have calculated the explicit solutions for the Grammian solutions of the model. We presented the dynamics of the solutions and by reducing the solutions also obtain the one-soliton solution for the semi-discrete model. This work can be extended in various interesting directions. For example, one can study discrete and semi-discrete versions of the multi-component GLHM model as well as their multi-soliton solutions. It would also be interesting to study discrete rogue and hump wave solutions for the GLHM model.