Zeeshan Amjadand Bushra Haider

Department of Physics, University of the Punjab, Quaid-e-Azam Campus, Lahore-54590, Pakistan

Abstract In this paper,we study the discrete Darboux and standard binary Darboux transformation for the generalized lattice Heisenberg magnet model.We calculate the quasi-Grammian solutions by the iteration of standard binary Darboux transformation.Furthermore, we derive the explicit matrix solutions for the binary Darboux matrix and then reduce them to the elementary Darboux matrix and plot the dynamics of solutions.

Keywords: quasi-Grammian, discrete integrable systems, binary Darboux transformation

1.Introduction

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,B?cklund transformation, Darboux transformation and other solutiongenerating 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, B?cklund 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 B?cklund 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 groupSU(2) transformation and present the solutions,which also include the soliton solution.

2.Lax pair

The Lax pair of the GLHM model is given as,having matricesAnandBngiven by,

where the matrix functionUn≡Un(t) take values from Lie group G in the Lie algebragand Ψn≡Ψn(λ) is anN×Neigen matrix, which depends upon variablenwritten in subscripts defined over a lattice.The matrix functionUnis subjected to constraints given byUn2=IandJnUn=Un?1Jn,which also impliesJnAn=An?1Jn.The compatibility condition dAn/dt+AnBn?Bn+1An=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 Δnfn=fn+1?fn.ForN=2, we have the simplest 2×2 case of Lie groupSU(2), for which the matrixUnis expressed asUn=Unaσa, where σaare the familiar Pauli matrices and the constraint on the matrixUnbecomesUn2=I.Iis the 2×2 identity matrix.We substitute 2 (Un+Un-1)-1=(Un+Un-1)/(1+UnUn-1)in (2.6) and are able to express the equation of motion in vector notation as,

3.Discrete Darboux transformation

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 theN×NDarboux matrixDn(λ) to calculate the soliton solutions.The Darboux matrix transforms the matrix solution from the spaceWto new space, i.e.

The one-fold Darboux transformation on matrix solution Ψnis defined as,

whereDn(λ) is the Darboux matrix.The new transformed solution Ψn[1] satisfies the following Lax pair (2.1) as,

havingAn[1] andBn[1] as,

whereIis the identity matrix.In order to obtain the Darboux transformation on matrix solutionUn[1], we define the Darboux matrix as,

whereIis theN×Nidentity matrix andQnis the auxiliary matrix ofN×Norder, which is yet to be found.The choice forQnisQn=HnΛ-1Hn-1, whereHnis the distinct matrix solution of the Lax pair (2.1) having orderN×N,which can be obtained by usingi-eigenvector functions Ψ(λi)∣σ〉ievaluated at λi, i=1,…,N, whereas matrix Λ is a diagonal matrix of orderN×Nhaving eigenvalues λ1, λ2,…,λN.Therefore, matrixHncan 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 asUnin equation(2.2), provided that matrixQnfulfills the following conditions:

Proof.The relation between the Darboux transformed solutionUn[ 1]and the untransformed solutionUnis developed and defined in equation (3.10).We then have to show that the choice of matrixQn=HnΛ-1Hn-1satisfies the condition (3.11), i.e.

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 asJnin equation(2.2), provided that matrixQnfulfills the following conditions:

Proof.The relation between the Darboux transformed solutionJn[1 ]and the untransformed solutionJnis developed and defined in equation(3.12).We then have to show that the choice of matrixQn=HnΛ-1Hn-1satisfies the condition(3.13).For this, we operateon matrix(I-Qn2)as,

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

Remark 1.Thus,the matrixQn=HnΛ-1Hn-1is a good choice,which satisfies the conditions imposed by the Darboux transformation.Thus, the Darboux transformation preserves the system,i.e.ifΨn,UnandJn,respectively,are the solutions of the linear system(2.1)and(2.2)and the equation of motion(2.4),then Ψn[ 1] ,Un[ 1]andWn[ 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 Ψnas,

For the next iteration of Darboux transformation, takeQn,1andQn,2as the two particular solutions of the Lax pair (3.3)and (3.4) at Λ = Λ-11and Λ = Λ-21, respectively.The twofold Darboux transformation on Ψn[1] is defined 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.1For a general quasideterminant expanded about N×N matrix D,we haveFrom the noncommutative Jacobi identity,we obtain the homological relationwhere O and I denote the null and identity matrices, respectively.Similarly, theK-fold Darboux transformation is given by,

The expression (3.10) can then be expressed as,

The result can be generalized toK-times Darboux transformation as,

The expressions given by equations (3.17) and (3.19) are theKth 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,

withAn?andBn?given by,

where η is a spectral parameter and Φnis an invertibleN×Nmatrix field in the adjoint spaceW?.The Darboux matrixDn(η)transforms the matrix solution Φnin spaceW?to a new matrix solutionini.e.

Based upon the above facts, we can write Darboux transformation Φnas,

whereSnis theN×Nmatrix that is to be determined andIisN×Nidentity matrix.The covariance of the Lax pair under the Darboux transformation requires that the new solutionsatisfies the Lax pair (3.20) given by,

By operating the Darboux transformation(3.23)on(3.24),we obtain the Darboux transformed matrix functionsUn?andJn?as,

The matrixSncan be constructed from the eigen matrices of the Lax pair and we takeSnto beSn=MnΞ-1Mn-1, where Ξ=diag(η1,…,ηn) is the eigenvalue matrix.The particular matrix solutionMnof the Lax pair (3.20) is an invertibleN×Nmatrix, which is given by,

Each column∣φi〉n= φn(ηi)∣ei〉inMnis a column solution of the Lax pair (3.20).TheK-fold Darboux transformation on the matrix solution and matrix function Φn,?Uncan be written as,

Similarly, the quasideterminant ofU?n[K] is,

Equations (3.27) and (3.28) are theKth quasideterminant solutions of the GLHM model for the adjoint space.

4.Standard binary Darboux transformation

In order to define the binary Darboux transformation, we consider the hat space, which is a copied version of direct spaceW, so the corresponding solutions are∈.Therefore, the equation of motion and the compatibility condition will have the identical form as that for the direct space given by,

The specific solutions for the direct and adjoint spaces areHnandSn,respectively.Thus,the corresponding solutions forareAlso assuming thatwe can then write the transformation as,

Since φn∈Wn?, we have,

Also fromDn?(λ)( i(Hn))= 0, we obtain i (Hn)=Mn(-1)?and similarlyTherefore, from the above equations we can write,

and

where the algebraic potential Δnis defined as,

Similarly, for the adjoint space matrixis written as,

where,

By writing equations (4.7) and (4.9) in matrix form for the solutionsHnandMn,we obtain the condition on Δnwhich is given by,

where the Δnmatrix 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(λ) on equation(4.1),the equation must be covariant, i.e.

The Darboux transformation on matrix fieldandin hat spaceis,

We then define the standard binary Darboux transformation from equation (4.14), which relatesandas,

which implies that,

We then operate the standard binary Darboux transformation on matrix solution Ψnas,

By using the expression (4.6) in the above equation, we obtain,

By substituting the expression of Δn(Hn, Φn)Λ?1from 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 Ψn.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 spacewe obtain,

The standard binary Darboux transformation on the solution of GLHM modelUn, is given by,

This is in fact a product of quasideterminants, i.e.

This expression can be further simplified when we introduce potential Δninstead 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 theKth iteration of Ψnthrough the iteration of binary Darboux transformation given by,

TheKth iteration for the adjoint binary Darboux transformation is given by,Similarly,[K]can be written as,

Figure 1.Dynamics of The Darboux transformation on matrix field ,11.

Similarly,using the iteration process we can calculate the quasideterminant solutions for

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.

5.Explicit solutions

In this section, we consider the GLHM model based on the Lie groupSU(2)and obtain the soliton solutions by using the binary Darboux transformation.To obtain an explicit expression for the soliton solution in the generalN×Ncase,we take the seed solution,

whereciare real constants and TrUn=0.A trivial calculation then yields a matrix solution Ψnof the Lax pair(2.1)with the form,

where,

Figure 2.Dynamics of .

Figure 3.Dynamics of Un+ for different values of c.

are respectivelyp×pand(N?p)×(N?p)matrices.Here,nin the subscript is a discrete index.We then take the seed solution for the caseN=2, which is given as,

Thus,the solution of the linear system(2.1)can be expressed as,

Figure 4.Dynamics of Un -: for different values of c.

Figure 5.Semi-discrete one-soliton solution Un+.

Figure 6.Semi-discrete one-soliton solution Un-.

where,

The particular matrix solutionHnof the direct Lax pair can be written by using the above equation (5.2) as,

The expression forQn=HnΛ-1Hn-1by usingbecomes,

where,

Similarly, the particular matrix solutionMnfor adjoint space can be written as,

where,

In order to obtain the expression for, we start from the definition of Δn(Hn,Mn) 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.

5.1.Reduction

Thus, we can write,

Therefore, the solution of matrix functionby using expressions (5.1) and (4.12) can be written as,

where,

The expressions (5.14) are depicted in figures 3 and 4.

From(5.15),it can be seen thatU?[1]=?U[1]and Tr(U[1])=0.Therefore,it can be said that the above expression(5.15)is an explicit equation based onSU(2)one-soliton solutions of the GLHM model.

We can obtain solutions in the form of direct and adjoint space parameters, by using the standard binary Darboux transformation, which are different to those obtained by elementary Darboux transformation.In addition, we reduce the solutions into the elementary Darboux transformation solutions,which is the advantage of binary Darboux transformation.

6.Conclusion

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 multisoliton solutions.It would also be interesting to study discrete rogue and hump wave solutions for the GLHM model.

Declaration of competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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