Positive and Negative Integrable Hierarchies: Bi-Hamiltonian Structure and Darboux–Bäcklund Transformation

Zhang, Ning; Xu, Xi-Xiang

doi:https://doi.org/10.1155/2020/5363952

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 5363952 | https://doi.org/10.1155/2020/5363952

Positive and Negative Integrable Hierarchies: Bi-Hamiltonian Structure and Darboux–Bäcklund Transformation

Ning Zhang¹and Xi-Xiang Xu²

Academic Editor: Irfan Kaymaz

Received21 May 2020

Accepted14 Dec 2020

Published30 Dec 2020

Abstract

Two integrable hierarchies are derived from a novel discrete matrix spectral problem by discrete zero curvature equations. They correspond, respectively, to positive power and negative power expansions of Lax operators with respect to the spectral parameter. The bi-Hamiltonian structures of obtained hierarchies are established by a pair of Hamiltonian operators through discrete trace identity. The Liouville integrability of the obtained hierarchies is proved. Through a gauge transformation of the Lax pair, a Darboux–Bäcklund transformation is constructed for the first nonlinear different-difference equation in the negative hierarchy. Ultimately, applying the obtained Darboux–Bäcklund transformation, two exact solutions are given by means of mathematical software.

1. Introduction

It is well known that the study of nonlinear integrable differential-difference equations (NIDDEs) has attracted much attention in recent decades [1–14]. Many problems in mathematical physics may be modeled by NIDDEs. Up to now, many important NIDDEs have been presented such as the Ablowitz–Ladik lattice [1], the Toda lattice [2], the relativistic Toda lattice [3], the modified Toda lattice [4, 5], the Merola–Ragnisco–Tu lattice [6], and the deformed reduced semidiscrete Kaup–Newell lattice [7–14]. Now, finding new NIDDEs, in lattice soliton theory, is still an important and complicated task. In general, we choose an appropriate discrete matrix spectral problem and a list of auxiliary spectral problems:where for a lattice function , the shift operator and the inverse of are defined by is a square matrix, is a list of the same order square matrices of , is a potential vector function, is the eigenfunction vector, is the spectral parameter, and . The integrability condition of (1) is , and it is equivalent to

Here, (3) is called a discrete zero curvature equation. Usually, (3) determines a hierarchy of NIDDEs (or lattice soliton equations):

One of the important problems in the lattice soliton theory is to search for a Hamiltonian operator and a hierarchy of conserved densities so that (4) has the following Hamiltonian structures:where the Hamiltonian functionals and the variational derivative . Furthermore, if there is another operator so that and form a pair of Hamiltonian operators andthen the integrable hierarchy (4) possess a bi-Hamiltonian structure. According to the theory of Hamiltonian operators, if is reversible, then the hierarchy (4) is integrable in Liouville sense ([4, 13] and their references). As is known to all, staring from a continuous matrix spectral problem, we only derive one integrable hierarchy, but for some suitable discrete matrix spectral problems, we can get two hierarchies of the NIDDEs [13]. In this paper, we are going to present two integrable hierarchies from a discrete matrix spectral problem. They, respectively, apply positive power and negative power expansions of Lax operators with respect to the spectral parameter. Theory is, respectively, called positive and negative integrable hierarchies. Moreover, as is known to all, Bäcklund transformation is a powerful method to obtain exact solutions of NIDDEs [15, 16]. This transformation is a relation between the new solution and the old solution of the NIDDEs. Based on a known solution, applying this transformation, a new solution may be derived. According to [15], Bäcklund transformation is usually divided into three types: the Wahlquist–Estabrook (WE) type [16, 17], the Hirota type [18], and the Darboux–Bäcklund type [19–22]. In the Darboux–Bäcklund type, the Lax pair plays a key role. A gauge transformation of the Lax pairs is called a Darboux–Bäcklund transformation if it transforms the Lax pair into another Lax pair of the same form.

This paper is organized as follows. In Sections 2 and 3, we introduce a novel discrete spectral problem:where is the eigenfunction vector, is the potential vector, and and depend on integer and real . Staring from spectral problem (7), positive and negative integrable hierarchies of NIDDEs are, respectively, presented by discrete zero curvature equations. Then, the Hamiltonian structure and bi-Hamiltonian structure of the obtained hierarchies are established by means of the discrete trace identity [11]. Afterwards, infinitely many common commuting conserved functionals of the obtained positive hierarchy are worked out. The Liouville integrability of the obtained positive hierarchy is proved. For the obtained negative integrable hierarchy, the same results can be similarly obtained. In Section 4, a Darboux–Bäcklund transformation is established though the gauge transformation of the Lax pair for the first NIDDE in the negative integrable hierarchy. In Section 5, using obtained Darboux–Bäcklund transformation, two exact solutions are given with the help of the mathematical software “Mathematica.” Finally, in Section 6, there will be some conclusions and remarks.

2. Positive Integrable Hierarchy and Its Bi-Hamiltonian Structure

Now, we want to deduce a hierarchy of NIDDEs associated with eigenvalue problem (7). For this purpose, we first solve the following stationary discrete zero curvature equation:

Let us set

We find that equation (8) implies

Substituting expansionsinto (10) and comparing each power of in equation (10), we obtain the initial conditions:and the recursion relations:

Proposition 1. We take thatthen , which are solved by equation (13), are all local, and they are just rational functions in the two dependent variables and .

Proof. According to second and third equations in (13), we get that and can be shown locally by means of , and . In order to derived from first equation in (13), we require to apply operator to solve the related difference equation. Next, we will show that may be also deduced through an algebraic method rather than by solving the difference equation. Based on (8), we obtain thatSo , is an arbitrary function of time variable only. Furthermore, we take that . Then, we obtain a recursion relation for :Therefore, can be derived locally by , and and then are all local; they are just rational functions in the two dependent variables and .
The proof is completed.
Specially, we haveLet us denoteBy means of (13), we getObviously, (19) is not compatible with . So, we choose a correction termand setWe consider the following auxiliary spectral problems associated with the spectral problem (7):Then, the compatibility condition of equations (7) and (22)is equivalent to the discrete zero curvature equationswhich give rise to the hierarchy of NLDDEs:

Remark 1. Owing to the entries in matrix only has nonnegative power powers of the eigenvalue , then the integrable hierarchy (25) is called a positive integrable hierarchy associated with the discrete matrix spectral problem (7).
When , (25) becomes a trivial linear system:When in (25), we obtain the first NIDDE in hierarchy (25):Furthermore, it is easy to derive the time part of the Lax pair of (27) isNext, we are going to establish the Hamiltonian structure for the hierarchy of NLDDE (25) by means of the discrete trace identity [11].
First, let us introduce some notions for further discussion. The Gateaux derivative and the inner product are defined, respectively, bywhere are demanded to be rapidly vanished at the infinity. The standard inner product of and in the Euclidean space is given by . Operator is defined by , and it is called the adjoint operator of . If an operator possesses the property , then is called to be a skew-symmetric. A linear operator is called a Hamiltonian operator if is a skew-symmetric operator and fulfills the Jacobi identity, i.e.,For a Hamiltonian operator , we may define a corresponding Poisson bracket [4]:According to [11], we writeand , where and are the same order square matrices. We haveHence,By virtue of the discrete trace identity [11],Substituting expansions into (35) and comparing the coefficients of , we arrive atWhen in equation (36), by means of a direct confirmation, we get that . Thus, equation (36) can be written asMoreover, we havewhereFurthermore, we can getwhere andIn above matrix, .

Proposition 2. For all values of two arbitrary constants and ,is a Hamiltonian operator.
Expressly, constitute a pair of Hamiltonian operators.

Proof. Obviously, the operator is a skew-symmetric operator, i.e., . Furthermore, by a direct and tedious calculation, we can prove that the operator fulfills the Jacobi identity (30).
The proposition is proved.
So we obtain the following proposition.

Proposition 3. Equation (25) possesses the following Hamiltonian structure:where .

According to equation (13), we get the recursion relation:

Furthermore, we have

That is, (25) is a hierarchy of discrete bi-Hamiltonian systems.

Using the operator , the positive integrable hierarchy (25) can be written as follows:

Obviously, the integrable NLDDE (27) possesses Hamiltonian structure:

Next, we prove the Liouville integrability of the discrete bi-Hamiltonian systems (25). It is crucial to make known the existence of infinite involutive conserved functionals.

Proposition 4. are conserved functionals of the whole family (25) or (45). And they are in involution in pairs with respect to the Poisson bracket (31).

Proof. Though a direct calculation, we havenamely,Therefore,Repeating the above argument, we can obtainBy equations (51) and (52), we haveThe proposition is proved.
Based on (45) and the Propositions 3 and 4, we can obtain the following theorem.

Theorem 1. The integrable NLDDE in hierarchy (25) is all Liouville integrable discrete bi-Hamiltonian systems.

3. Negative Integrable Lattice Hierarchy and Its Bi-Hamiltonian Structure

In this section, we would like to derive the negative integrable hierarchy associated with matrix spectral problem (7). To this end, we first consider the following stationary discrete zero curvature equation:with

On the basis of equation (54), we arrive at

Here, we expand by the nonnegative power power of :

Substituting the above expansions into (56), we get the following initial conditions:and recursion relations

If the initial values are chosen asthen we get that

Similar to Proposition 1, we can obtain that are all local, and they are just polynomial functions in the two dependent variables and , and may be deduced through an algebraic method rather than by solving the difference equation.

The first few terms are given by

Set

At this point,

It is obvious that (64) is not also compatible with . So, we choose the following correction term:

Then, we introduce auxiliary matrix spectral problem:

Through a direct calculation, we obtain thatit is equivalent to

Remark 2. Because the entries in matrix only have negative powers of the eigenvalue , then the integrable hierarchy (68) is called a negative integrable hierarchy associated with the discrete matrix spectral problem (7).
When , (68) becomesIf we set , (69) is reduced to the well-known Volterra lattice ; namely, (69) is a generalized Volterra lattice.
In next section, we are going to establish a Darboux–Bcklund transformation of (69). It is easy to get the time part of the Lax pair of (69) isIn the discrete variational identity (35), we replace with ; at this point, the following equations hold:where .
Substituting expansionsinto (71), we getwhere . There is the recursion relation as follows:where .
Based on (56), we getThen, we haveThus, the integrable hierarchy (68) has a bi-Hamiltonian structure (76). Furthermore, similar to integrable hierarchy (25), we can prove that integrable hierarchy (68) is also Liouville integrable.
With the help of the operator , the negative integrable hierarchy (68) can be written as follows:

4. Darboux–Bäcklund Transformation

We introduce a gauge transformation of spectral problem (7):

Under this transformation, two spectral problems (7) and (70) becomewith

Here, we supposewhere are undetermined functions of variables and and . Next, we are going to solve such that and in equation (80) are provided with the same form with and , i.e.,

For two different reals , we can get that are two real linear independent solutions of equations (7) and (70):

Let , and we get

We consider

In the above equation, set , , are nonzero constants.

Solving equation (85) for , we obtain that

In equation (86),

Through direct calculation, we have

Proposition 5. The matrix defined by (80) has the same form as in equation (7), and the original potentials are changed into new potentials by means of

Proof. We know that is the adjoint matrix of ; from equations (7) and (87), we obtain thatwithFrom equations (90) and (91), we arrive atAs a result, we havewhereThen, we find that and are two 3th-order polynomial in and and are two 2th-order polynomial in . By a tedious but direct computation or by a mathematical software, we get that are two roots of . Thus, we may assumewithwhere are all independent of . Thus, we obtainBy comparing the coefficients of , in both sides of equation (97), we get thatHence, we obtain that .
The proof is finished.

Proposition 6. The matrix given in (80) has the same form as in (70) by means of the transformation (89).

Proof. Let us denotewhereBased on (70) and (87), we getThrough tediously long calculation, we can find that . So, we havewhereand are all independent of .
By means of equation (102), we obtainComparing the coefficients of in (104), we haveThus,The proof is finished.
The transformations (78) and (89), namely, from to constitute a Darboux–Bäcklund transformation of the NIDDE (69).
In conclusion, according to the Propositions 5 and 6, we have the theorem.

Theorem 2. Every solution of the NLDDE (69) is changed into a new solution under the Darboux–Bäcklund transformation (89).

5. Exact Solutions

Next, we will use the Darboux–Bäcklund transformation (89) to find two solutions of equation (69).

First, we consider a seed solution of (69) (a simple special solution) . Substituting this solution into the corresponding Lax pair (7) and (70), we have

Solving above two equations, we arrive at the solutions

Then, we obtain

Using the Darboux–Bcklund transformation (89) and with the help of mathematical software ”Mathematica,” we obtain an exact solution of (69):

Then, it easy to verify that is another seed solution of (69). Substituting it into the corresponding Lax pair, it is found that

Set

By means of mathematical software “Mathematica,” we may get that two real linear independent solutions as follows:where

Based on (87), we have

Though the Darboux–Bäcklund transformation (89), we arrive at another exact solution of (69):

6. Conclusions and Remarks

In this paper, we have deduced two hierarchies of NIDDEs from a discrete matrix spectral problem by the discrete zero curvature equation. The obtained hierarchies, respectively, work in concert with positive power and negative power expansions of Lax operators with respect to the spectral parameter. Two bi-Hamiltonian forms for the obtained integrable hierarchies are given by the discrete trace identity. And then, the Liouville integrability of the obtained hierarchies is demonstrated. Furthermore, by the aid of a gauge transformation of the Lax pair, a Darboux–Bäcklund transformation for the first NIDDE in the negative integrable hierarchy was presented. Applying the obtained Darboux–Bäcklund transformation and “Mathematica,” we get two exact solutions. These solutions also are called one-fold solutions. This Darboux–Bäcklund transformation is continuously done N times, then N-fold solution of (69) can be derived. Besides, we can get the Darboux–Bäcklund transformation of the first NIDDE (27) in the positive hierarchy in a similar way. Recently, in the soliton theory, some new types of explicit solutions for the continuous soliton equations have been found, for instance, abundant lump solutions and interaction solutions [23–26]. For the NIDDE (69), explicit solutions of these types can also be researched. These results will appear in later papers.

In addition, many interesting problems deserve further investigation for the NIDDEs in the obtained hierarchies (25) and (68), such as symmetries constraint, integrable coupling systems by semidirect sums of Lie algebra, symmetries, and master symmetries.

Data Availability

The data used to support the findings of this study are available from the corresponding author on reasonable request.

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 Youth Science Foundation Project of China (Grant no. 11805114).

References

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, England, 1991.
M. Toda, Theory of Nonlinear Lattice, Springer-Verlag, Berlin, Germany, 2 edition, 1989.
H. Flaschka, “On the Toda lattice. II: inverse-scattering solution,” Progress of Theoretical Physics, vol. 51, no. 3, pp. 703–716, 1974.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar
X.-X. Xu, H.-X. Yang, and Y.-P. Sun, “Darboux transformation of the modified Toda lattice equation,” Modern Physics Letters B, vol. 20, no. 11, pp. 641–648, 2006.
View at: Publisher Site | Google Scholar
I. Merola, O. Ragnisco, and T. Gui-Zhang, “A novel hierarchy of integrable lattices,” Inverse Problems, vol. 10, no. 6, pp. 1315–1334, 1994.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar
X. Geng, H. H. Dai, and C. Cao, “Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications,” Journal of Mathematical Physics, vol. 44, no. 10, pp. 4573–4588, 2003.
View at: Publisher Site | Google Scholar
T. Tsuchida, “Integrable discretizations of derivative nonlinear Schr dinger equations,” Journal of Physics A: Mathematical and General, vol. 35, no. 36, pp. 7827–7847, 2002.
View at: Publisher Site | Google Scholar
A. Pickering and Z.-N. Zhu, “New integrable lattice hierarchies,” Physics Letters A, vol. 349, no. 6, pp. 439–445, 2006.
View at: Publisher Site | Google Scholar
G. Z. Tu, “A trace identity and its applications to theory of discrete integrable systems,” Journal of Physics A: Mathematical and General, vol. 23, pp. 3903–3922, 1990.
View at: Google Scholar
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.
View at: Publisher Site | Google Scholar
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.
View at: Publisher Site | Google Scholar
M. Blaszak and K. Marciniak, “R-matrix approach to lattice integrable systems,” Journal of Mathematical Physics, vol. 35, pp. 4661–4682, 1994.
View at: Google Scholar
D. Y. Chen, “Backlund transformation and soliton solutions,” Journal for Research in Mathematics Education, vol. 25, pp. 479–488, 2005.
View at: Google Scholar
H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the korteweg-de Vries equation,” Physical Review Letters, vol. 31, no. 23, pp. 1386–1390, 1973.
View at: Publisher Site | Google Scholar
V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin,Gemany, 1991.
R. Hirota, “A new form of backlund transformations and its relation to the inverse scattering problem,” Progress of Theoretical Physics, vol. 52, no. 5, pp. 1498–1512, 1974.
View at: Publisher Site | Google Scholar
C. H. Gu, H. S. Hu, and Z. X. Zhou, Soliton Theory and its Application, Zhejiang Science and Technology Publishing House, Berlin,Germany, 1995.
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, pp. 677–684, 1998.
View at: Publisher Site | Google Scholar
X.-X. Xu and Y.-P. Sun, “An integrable coupling hierarchy of Dirac integrable hierarchy, its Liouville integrability and Darboux transformation,” The Journal of Nonlinear Sciences and Applications, vol. 10, no. 06, pp. 3328–3343, 2017.
View at: Publisher Site | Google Scholar
R.-W. Lu, X.-X. Xu, and N. Zhang, “Construction of solutions for an integrable differential-difference equation by Darboux-Bäcklund transformation,” Applied Mathematics and Computation, vol. 361, pp. 389–397, 2019.
View at: Publisher Site | Google Scholar
W. X. Ma, “Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs,” Journal of Geometry and Physics, vol. 133, pp. 45–54, 2018.
View at: Publisher Site | Google Scholar
S.-T. Chen and W.-X. Ma, “Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation,” Frontiers of Mathematics in China, vol. 13, no. 3, pp. 525–534, 2018.
View at: Publisher Site | Google Scholar
W. X. Ma, J. Li, and C. M. Khalique, “A study on lump solutions to a generalized Hirota-Satsuma-Ito equation in (2+1)-dimensions,” Complexity, vol. 2018, Article ID 9059858, 7 pages, 2018.
View at: Publisher Site | Google Scholar
J.-Y. Yang, W.-X. Ma, and Z. Qin, “Lump and lump-soliton solutions to the $$(2+1)$$ ( 2 + 1 ) -dimensional Ito equation,” Analysis and Mathematical Physics, vol. 8, no. 3, pp. 427–436, 2018.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Ning Zhang and Xi-Xiang 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.

PDF Download Citation

Download other formats

Order printed copies

Views

378

Downloads

685

Citations