Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 4152917, 11 pages
https://doi.org/10.1155/2018/4152917

Research Article

A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation

Xi-Xiang Xu and Meng Xu

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Xi-Xiang Xu; xixiang_xu@sohu.com

Received 26 December 2017; Accepted 5 May 2018; Published 7 June 2018

Academic Editor: Josef Diblik

Copyright © 2018 Xi-Xiang Xu and Meng Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

An integrable family of the different-difference equations is derived from a discrete matrix spectral problem by the discrete zero curvature representation. Hamiltonian structure of obtained integrable family is established. Liouville integrability for the obtained family of discrete Hamiltonian systems is proved. Based on the gauge transformation between the Lax pair, a Darboux-Bäcklund transformation of the first nonlinear different-difference equation in obtained family is deduced. Using this Darboux-Bäcklund transformation, an exact solution is presented.

1. Introduction

During the past decades, study of the integrable different-difference equations (or integrable lattice equations) has received considerable attention. They play the important roles in mathematical physics, lattice soliton theory, cellular automata, and so on. Many nonlinear integrable different-difference equations have been proposed and discussed, such as Ablowitz-Ladik lattice [1], Toda lattice [2], relativistic Toda lattices in polynomial form and rational form [3, 4], modified Toda lattice [5, 6], Volterra (or Langmuir) lattice [7], deformed reduced semi-discrete Kaup-Newell lattice [8], Merola-Ragnisco-Tu lattice [9], and so forth [10–18]. As we know, searching for novel integrable nonlinear different-difference equations is still an important and very difficult research topic. In the lattice soliton theory, the discrete zero curvature representation is a significant way to derive the integrable different-difference equations [18].

Starting from a suitable matrix spectral problem

and a series of auxiliary spectral problems

where for a lattice function , the shift operator and the inverse of are defined by

and is eigenfunction vector, and is a potential vector.

A family of evolution different-difference equations

is called to be integrable in Lax sense (or lattice soliton equations), if it can be written as a compatibility condition of equations (1) and (2):

Moreover, in the lattice soliton theory, if we can seek out a Hamiltonian operator and a series of conserved functionals so that (4) may be represented as the following forms

where the Hamiltonian functionals , then Hamiltonian structure of family (4) is established. In lattice soliton theory, (6) is called discrete Hamiltonian systems. Further, if we can deduce infinitely many involutive conserved functionals of family (6), then Liouville integrability of the integrable lattice family (4) is proved.

In addition, it is well known that Darboux-Bäcklund transformation is an important and very effective method for solving integrable different-difference equations. This transformation is a formula between the new solution and the old solution of the different-difference equation. From a known solution, through this transformation, we can obtain another solution. According to [19], Bäcklund transformation are usually divided into three types: the Wahlquist-Estabrook (WE) type [20, 21], the Darboux-Bäcklund type [5, 6, 22, 23], and the Hirota type [24].

This paper is organized as follows. In Section 2, we introduce a novel discrete spectral problem

where is the spectral parameter and . It is easy to see that the matrix spectral problem (7) is equivalent to the following eigenvalue problem:

the eigenfunction . Based on this discrete matrix spectral problem, we derive a family of integrable different-difference equations through the discrete zero curvature representation. Then, we establish the Hamiltonian structure of the obtained family by means of the discrete trace identity [18]. Afterwards, infinitely many common commuting conserved functionals of the obtained family are worked out. This guarantees Liouville integrability of the obtained family. In Section 3, we would like to derive a Bäcklund transformation of Darboux type (or Darboux-Bäcklund transformation) of the first nonlinear integrable different-difference equation in the obtained family; this transformation is constructed by means of the gauge transformation of Lax pair of the spectral problem, as application of Darboux-Bäcklund transformation, an exact solution is given. Some conclusions and remarks are given in the final section.

2. An Integrable Different-Difference Family and Its Hamiltonian Structure

We first solve the stationary discrete zero curvature equation

with

Equation (9) implies

Substituting Laurent series expansions

into (11), we obtain the initial conditions:

and the recursion relations:

We choose the initial values satisfying the above initial conditions,

and require selecting zero constant for the inverse operation of the difference operator in computing , then the recursion relation (14) uniquely determines . In addition, we have the following assertion.

Proposition 1. may be deduced through an algebraic method rather than by solving the difference equation.

Proof. From (9), we know that

Here is an arbitrary function of time variable only. Further, we choose Then, we get a recursion relation for . Thus, are all local, and they are just the rational function in two dependent variables .
The proof is completed.

The first few terms are given by

Let us denote

On the basis of the recursion relations (14), we have

It is obvious that (19) is not compatible with . Therefore, we choose the following modification term:

Then we introduce auxiliary matrix spectral problem

Let the time evolution of the eigenfunction of the spectral problem (7) obey the differential equation

Then the compatibility conditions of (7) and (22) are

It implies the family of integrable (in Lax sense) different-difference equations.

When , (24) becomes a trivial linear system

When in (24), we obtain the first nontrivial integrable lattice equation

In Section 3, we are going to construct its Darboux-Bäcklund transformation.

Now let us introduce some concepts for further discussion. The Gateaux derivative, the variational derivative, and the inner product are defined, respectively, by

are required to be rapidly vanished at the infinity, and denotes the standard inner product of and in the Euclidean space . Operator is defined by ; it is called the adjoint operator of . If an operator has the property , then is called to be skew-symmetric. An operator is called a Hamiltonian operator, if is a skew-symmetric operator satisfying the Jacobi identity, i.e.,

Based on a given Hamiltonian operator , we can define the corresponding Poisson bracket [10, 11, 18]

Following [18], we set

and , where and are some order square matrices. It is easy to calculate that

Hence

By virtue of the discrete trace identity [18]

Substituting expansions into (33) and comparing the coefficients of in (33), we obtain

When in (34), through a direct calculation, we find that . Therefore, (34) can be written as

Set

We have

Set

We can obtain

Evidently, the operator is a skew-symmetric operator, i.e., . In addition, through a direct calculation, we can prove that the operator satisfies the Jacobi identity (28). Thus, we obtain the following assertion.

Proposition 2. is a discrete Hamiltonian operator.
Consequently, (22) have Hamiltonian structures

In particular, the different-difference equation (23) possesses the Hamiltonian structure

Following (14) and (35), we have the following recursion relation:

where

Moreover, we have

with

It is easy to verify that is a skew-symmetric operator. Namely,

With the help of the operator , (40) may be written in the following form:

Furthermore, on the basis of [4, 16], we may obtain a recursion operator

Next, we prove Liouville integrability of the discrete Hamiltonian systems (40). It is crucial to show the existence of infinite involutive conserved functionals.

Proposition 3. are conserved functionals of the whole family (24) or (40). And they are in involution in pairs with respect to the Poisson bracket (29).

Proof. Due to (47), we know that is a skew-symmetric operator, and a direct calculation shows

namely,

Therefore

Repeating the above argument, we can obtain

Then combining (52) with (53) leads to

and

The proof is completed.

Remark 4. According to the above proposition, we can get that (24) is not only Lax integrable but also Liouville integrable. Based on (40) and Proposition 2, we can obtain the following theorem.

Theorem 5. The integrable different-difference equations in family (24) are all Liouville integrable discrete Hamiltonian systems.

3. Darboux-Bäcklund Transformation and Exact Solution

Next we are going to establish a Darboux-Bäcklund transformation of (26).

When in (23), we obtain the time part of the Lax pair of (26)

It is crucial to look for a suitable gauge transformation of a matrix spectral problem; it can transform the matrix spectral problem into another spectral problem of the same form [5, 6, 8, 23]. We introduce the following gauge transformation:

which can transform two spectral problems (7) and (56) into

with

We suppose that has the following form:

In (61), , , , and are undetermined functions of variables and . Now we would like to construct such that and in (60) have the same form with and , respectively.

Let be two real linear independent solutions of (7) and (56). Let

From equation , we obtain that

We assume that and are four roots of . When , two columns of (63) are linear dependent. Without loss of generality, there exist two nonzero constants and , which satisfy

where

Set

Solving (64), we have

In (66), are suitably chosen such that all the denominations in (65) and (66) are nonzero. Through a direct but tedious calculation, we can get that are four roots of , and then

From (65), we get

where

Based on (65) and (66), we can get the expressions of .

Proposition 6. The matrix defined by (60) has the same form as , that is,

where the relations between old potentials and and new potentials and are given by

Proof. One has

with

According to (66) and (69), we can find that

Thus, we have . Moreover, it is easy to see that and are five-order polynomial in , and is a six-order polynomial in . Through a tedious but direct computation or by computer algebra system (for instance, Mathematica, Maple), we can obtain that

are all equal zero. Therefore, we have

with

where are all independent of . We know that . Hence we obtain

Equating the coefficients of in (78), we have

The proof is completed.

Proposition 7. The matrix defined by (60) has the same form as in (56) under the transformation (71), i.e.,

Proof. Let

Here

It is easy to get that and are six-order polynomials in ; and are five-order polynomials in . According to (56) and (65), we have

Through lengthy but direct calculation, we can verify that

Therefore, the following equation is derived:

with

where are all independent of . From (85), we get

Comparing the coefficients of in (87), we have

The proof is completed.

Theorem 8. Equation (71) is a Darboux-Bäcklund transformation of (26). That is, each solution of the integrable lattice system (26) is mapped into its new solution under transformation (71).

Next, using obtained Darboux-Bäcklund transformation (71), we derive an exact solution of (26).

First, it is easy to find that constitute a trivial solution of (71). Substituting this solution into the corresponding Lax pair, we get

Solving the above two equations, we get two real linear independent solutions:

where

From (65), we have

In (92), ,

By means of the Darboux-Bäcklund transformation (71), we deduce a new exact solution of (26).

This transformation can be done continually. Thus, we can obtain a series of exact solutions of (26).

4. Conclusions and Remarks

In this work, based on a Lax pair, we have deduced a novel family of integrable different-difference equations using the discrete zero curvature equation and established the Hamiltonian structure of the obtained integrable family of different-difference equations in virtue of the discrete trace identity. And then the Liouville integrability of the obtained family is demonstrated. With the help of a gauge transformation of the Lax pair, a Darboux-Bäcklund transformation for the first nonlinear different-difference equation in the obtained family is presented, and using the obtained Darboux-Bäcklund transformation, an exact solution is derived. It is worth noting that in (61) we can also study the generalized form of

with

In addition, many interesting problems deserve further investigation for obtained integrable family, for example, symmetry constraint [7], symmetries and master symmetries [13], and integrable coupling systems [25, 26].

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Science and Technology Plan Projects of the Educational Department of Shandong Province of China (Grant no. J08LI08).

References

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, NY, USA, 1991.

M. Toda, Theory of Nonlinear Lattices, vol. 20, Springer-Verlag, Berlin, Germany, 2nd edition, 1989.

H. Flaschka, “On the Toda lattice. II. Inverse-scattering solution,” Progress of Theoretical and Experimental Physics, vol. 51, no. 3, pp. 703–716, 1974.

W.-X. Ma and X.-X. Xu, “Positive and negative hierarchies of integrable lattice models associated with a Hamiltonian pair,” International Journal of Theoretical Physics, vol. 43, no. 1, pp. 219–235, 2004.

X.-X. Xu, H.-X. Yang, and Y.-P. Sun, “Darboux transformation of the modified Toda lattice equation,” Modern Physics Letters B. Condensed Matter Physics, Statistical Physics, Applied Physics, vol. 20, no. 11, pp. 641–648, 2006.

A. Pickering and Z. N. Zhu, “Darboux-Backlund transformation and explicit solutions to a hybrid lattice of the relativistic Toda lattice and the modified Toda lattice,” Physics Letters A, vol. 378, no. 21, pp. 1510–1513, 2014.

W.-X. Ma and X. Geng, “Backlund transformations of soliton systems from symmetry constraints,” CRM Proceedings and Lecture Notes, vol. 29, pp. 313–323, 2001.

X.-X. Xu, “A deformed reduced semi-discrete Kaup-Newell equation, the related integrable family and Darboux transformation,” Applied Mathematics and Computation, vol. 251, pp. 275–283, 2015.

I. Merola, O. Ragnisco, and G. Z. Tu, “A novel hierarchy of integrable lattices,” Inverse Problems, vol. 10, no. 6, pp. 1315–1334, 1994.

W.-X. Ma, “A discrete variational identity on semi-direct sums of Lie algebras,” Journal of Physics A: Mathematical and General, vol. 40, no. 50, pp. 15055–15069, 2007.

W.-X. Ma and X.-X. Xu, “A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations,” Journal of Physics A: Mathematical and General, vol. 37, no. 4, pp. 1323–1336, 2004.

W.-X. Ma, X.-X. Xu, and Y. Zhang, “Semidirect sums of Lie algebras and discrete integrable couplings,” Journal of Mathematical Physics, vol. 47, no. 5, Article ID 053501, 16 pages, 2006.

B. Fuchssteiner and W.-X. Ma, “An approach to master symmetries of lattice equations,” in Symmetries and Integrability of Difference Equations (Canterbury, 1996), P. A. Clarkson and F. W. Nijhoff, Eds., vol. 255, pp. 247–260, Cambridge University Press, Cambridge, England, 1999.

T. Tsuchida, “Integrable discretizations of derivative nonlinear Schrodinger equations,” Journal of Physics A: Mathematical and General, vol. 35, no. 36, pp. 7827–7847, 2002.

M. Blaszak and K. Marciniak, “R-matrix approach to lattice integrable systems,” Journal of Mathematical Physics, vol. 35, no. 9, pp. 4661–4682, 1994.

F. Khanizadeh, A. V. Mikhailov, and J. P. Wang, “Darboux transformations and recursion operators for differential-difference equations,” Theoretical and Mathematical Physics, vol. 177, no. 3, pp. 1606–1654, 2013.

D. Zhang and D. Chen, “Hamiltonian structure of discrete soliton systems,” Journal of Physics A: Mathematical and General, vol. 35, no. 33, pp. 7225–7241, 2002.

G. Z. Tu, “A trace identity and its applications to the theory of discrete integrable systems,” Journal of Physics A: Mathematical and General, vol. 23, no. 17, pp. 3903–3922, 1990.

D. Y. Chen, “Backlund transformation and N soliton solutions,” Formerly Journal of Mathematical Research and Exposition (JMRE), vol. 25, no. 3, pp. 479–488, 2005 (Chinese).

H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the Korteweg-de Vries equation,” Physical Review Letters, vol. 31, pp. 1386–1390, 1973.

V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Gemany, 1991.

C. H. Gu, H. S. Hu, and Z. X. Zhou, Soliton Theory and Its Application , Zhejiang Science and Technology Publishing House, Berlin, Germany, 1995 (Chinese).

Y. T. Wu and X. G. Geng, “A new hierarchy of integrable differential-difference equations and Darboux transformation,” Journal of Physics A: Mathematical and General, vol. 31, no. 38, pp. 677–684, 1998.

R. Hirota, “A new form of Bddotsacklund transformations and its relation to the inverse scattering problem,” Progress of Theoretical Physics, vol. 52, pp. 1498–1512, 1974.

X.-X. Xu and Y.-P. Sun, “An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation,” Journal of Nonlinear Sciences and Applications (JNSA), vol. 10, no. 6, pp. 3328–3343, 2017.

X.-X. Xu, “An integrable coupling hierarchy of the MKdV-integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 344–353, 2010.