/ / Article

Research Article | Open Access

Volume 2019 |Article ID 5984356 | https://doi.org/10.1155/2019/5984356

Qianqian Yang, Qiulan Zhao, Xinyue Li, "Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy", Complexity, vol. 2019, Article ID 5984356, 10 pages, 2019. https://doi.org/10.1155/2019/5984356

# Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Accepted28 May 2019
Published17 Jun 2019

#### Abstract

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

#### 1. Introduction

Investigation of nonlinear integrable lattice equations  is an active topic in the field of nonlinear science because its solutions can explain some physical phenomena . Researchers in soliton field have proposed many valuable nonlinear integrable lattice equations  until now, but there are still many integrable lattice equations that have not been discovered. Generally speaking, nonlinear integrable lattice equations are related to their matrix spectral problems, respectively. Therefore, finding suitable matrix spectral problems is of great significance for us to derive integrable lattice equations and investigate their explicitly exact solutions.

It is well known that it is not easy to get solutions for integrable lattice systems. Currently, there are several useful approaches which can help us obtain explicit solutions, for instance, the binary nonlinearization , Lie symmetry analysis [19, 20], and the Darboux transformation . The Darboux matix method is one of important and effective techniques to obtain explicitly exact solutions of integrable lattice equations .

In this paper, we will focus on this spectral problem:and its auxiliary problem,where , and are potentials, and is a spectral parameter independent of t. According to the above spectral problem, an integrable lattice hierarchy in Liouville sense is presented firstly. Then, we investigate its bi-Hamiltonian structures using the discrete trace identity  proposed by G. Z. Tu. Its infinitely many conservation laws are also presented. Finally, the explicitly exact solutions of the integrable lattice equation derived from this matrix spectral problem are discussed by DT method.

The structure of the article is as follows. In Section 2, we will obtain a new integrable hierarchy in Liouville sense and present its bi-Hamiltonian structure in accordance with the matrix spectral problem. In Section 3, we will discuss conservation laws of the integrable hierarchy proposed in Section 2. In Section 4, its N-fold DT will be constructed and its exact solutions will be obtained. In Section 5, some conclusions are given.

#### 2. Integrability and Bi-Hamiltonian Structures of a Lattice Hierarchy

Let us first introduce a shift operator , which obeys the following operations:where is a lattice function.

In order to obtain a hierarchy of integrable lattice equations, we assume thatBy solving the stationary discrete zero curvature equationthe following relationships will be derived:Substitutinginto (6) follows the recursion relations By calculation, we find and is a arbitrary constant. For simplicity of calculation, we take . Then, the recursion relations Equation (8) could uniquely determine , , , and . The first few quantities are given byand so on. Now, for any integer , we defineIn order to derive the associated integrable lattice hierarchy, we give a modification ,Lettingwe can obtain thatThen, the discrete zero-curvature equationis able to bring about the following integrable lattice hierarchy: and compose the Lax representation of the integrable lattice hierarchy Equation (15). When m=1, (15) is reduced to the following:in which is

Next, we try to construct the bi-Hamiltonian structures of (15). In order to achieve this goal, we denote and stands for the trace of the product of two square matrices of the same orders M and N. Then,The discrete trace identity  plays an important role in constructing Hamiltonian structures of integrable lattice equations and studying their integrability. Here, we try to construct Hamiltonian structure of (15) by using Substituting the expansions and into (19), we haveTo get the value of the constant , we simply set in (20) and then find . Therefore we have thatSo we obtain the desired Hamiltonian form of (15),whereIn (23), the operator is a Hamiltonian operator for the hierarchy Equation (15), and the operator is a recursion operator for the hierarchy Equation (15). These two operators and satisfy the following relationships:where stands for the formal conjugation of and

Remark. The hierarchy of lattice equations (15) is an integrable Hamiltonian system in Liouville sense.

The bi-Hamiltonian structure Equation (23) of the integrable lattice equation exists.

#### 3. Conservation Laws of (16)

Based on Lax representation of the hierarchy of integrable lattice equations (15), we can obtain its conversation laws. Substituting into the spectral problem (1), we havewhich demonstrates thatwhere we have .

Then, if we suppose that , the following two recursion relationships will be found. One isand the other isBased on (2) and (17), a direct calculation givesThen, we discuss conversation laws of the hierarchy of integrable lattice equations (15) according to the above two different recursion relationships, respectively. Substituting into (34), equating the coefficients of on both sides of the equation, we can achieve an infinite number of conversation laws of (16) based on the (32). The first two of them areSimilarly, we also have another infinite conservation laws of (16) based on (33). The first two of them are

#### 4. The N-Fold DT and Exact Solutions of Eq. (16)

If a integrable lattice equation with Lax pair and exists a gauge transformationwhere is a reversible matrix andThe new potentials and possess the same form with the old potentials and , respectively. The gauge transformation in (37) composes Darboux transformation of an integrable lattice equation with the transformation between old potential functions , and new potential functions   .

Based on (37) and (38), we are also able to obtain the relations, i.e.,

##### 4.1. The N-Fold DT of Eq. (16)

The selection of is very important in constructing the Darboux transformation. The solution of the soliton equation can be obtained much more easily by an appropriate Darboux matrix. For this integrable lattice equation (16), we take with and being all functions with respect to and and being a positive integer number. It is easy to find that is a (2N)-th order polynomial with respect to . If we assume that the 2N parameters () are the roots of the , can be expressed asSetting and being the column vectors of , we find that the column vectors of are linear dependent because of . Then for , according to (37), we havein whichand the parameters and are so chosen that the denominators in (43) are nonzero.

Theorem 1. The matrix defined by (38) has the same form as , i.e.,where the transformation from the old potentials and into the new potentials and is given by

Proof. Let andIt can be see that and are (2N-1)-th order polynomials with respect to , and and are (2N)-th order polynomials with respect to . From (42), we haveMoreover, (46) can be described astogether withwhere are roots of , is a 2-order square matrix, and are the functions with respect to n and t. Thus, (48) can be written asEquating the coefficients of and on both sides of (50), we findFrom (51), we can see that . The proof is completed.

Theorem 2. Under the transformation (37) and (45), the matrix defined by (38) has the same form as , i.e.,

Proof. Let andin which all are the functions of n and t. Through a series of calculations, we can figure out that , , and are (2N+1)-th order polynomials with respect to , and is (2N)-th order polynomial with respect to . We can verify from (47) that are the roots of . So, we knowtogether withwhere all the are the functions with respect to and . Thus, (54) can be rewritten asEquating the coefficients of and on both sides of (56), we find thatFrom (38) and (57), we are able to see that . The proof is completed.

##### 4.2. Exact Solutions of (16)

In this section, the explicit solutions of the integrable lattice equation (16) will be discussed under its Darboux transformation. We can find is a trivial solution of (16) easily. Starting from this trivial solution, the solutions of the Lax Pair Equations (1) and (17) can be obtained,in whichAccording to (43), we haveBy using of the Cramer rules, solving (42) followstogether withand is obtained from by replacing its (2N)-th column with , i.e.,By using (41), (45), and (62), the solution of (16) can be derived, and it is expressed asIn order to investigate the structure of the solutions (65) fully, we plot their structure figures as shown in Figures 14 when and .

(I) In particular, when , solving (42) leads towhereFigures 1 and 2 show the structures of the soliton solutions and in (65).

(II) When N=2, solving (42) leads towhereFigures 3 and 4 show the structures of the soliton solutions and in (68).

#### 5. Conclusions

In this article, we have derived an integrable lattice hierachy on the basis of a new matrix spectral problem. Then, some properties of this hierarchy have been shown, such as the Liouville integrability, the bi-Hamiltonian structures, and infinitely many conservation laws. Eventually, the Darboux transformation of the first integrable equation in this hierarchy has been constructed and the exact solutions of the integrable equation have been investigated via graphs. The binary nonlinearization  is also a useful method to investigate explicit solutions. We will investigate the binary nonlinearization with regard to (16) associated with spectral problems (1) and (2) in subsequent papers.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

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

#### Acknowledgments

The authors greatly appreciates Professor X. Y. Wen for his patient guidance and discussion. The work was supported by the Nature Science Foundation of China (No. 11701334) and the Science and Technology Plan Project of the Educational Department of Shandong Province of China (No. J16LI12).

1. M. A. Helal, “Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics,” Chaos, Solitons & Fractals, vol. 13, no. 9, pp. 1917–1929, 2002. View at: Publisher Site | Google Scholar | MathSciNet
2. Q.-L. Zhao and X.-Y. Li, “A Bargmann system and the involutive solutions associated with a new 4-order lattice hierarchy,” Analysis and Mathematical Physics, vol. 6, no. 3, pp. 237–254, 2016. View at: Publisher Site | Google Scholar | MathSciNet
3. X.-Y. Wen and Y.-T. Gao, “N-soliton solutions and elastic interaction of the coupled lattice soliton equations for nonlinear waves,” Applied Mathematics and Computation, vol. 219, no. 1, pp. 99–107, 2012. View at: Publisher Site | Google Scholar | MathSciNet
4. A. Kaldmäe, ü. Kotta, and M. Wadati, “On flatness of discrete-time nonlinear systems,” IFAC Proceedings Volumes, vol. 46, no. 23, pp. 588–593, 2013. View at: Google Scholar
5. A. Picozzi, J. Garnier, T. Hansson et al., “Optical wave turbulence: towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics,” Physics Reports, vol. 542, no. 1, pp. 1–132, 2014. View at: Publisher Site | Google Scholar | MathSciNet
6. X.-Y. Li, X.-X. Xu, and Q.-L. Zhao, “The integrable property of Lotka-Volterra type discrete nonlinear lattice soliton systems,” Modern Physics Letters B. Condensed Matter Physics, Statistical Physics, Atomic, Molecular and Optical Physics, vol. 22, no. 21, pp. 2007–2019, 2008. View at: Publisher Site | Google Scholar | MathSciNet
7. Y. Zhang, H. Dong, X. Zhang, and H. Yang, “Rational solutions and lump solutions to the generalized (3 + 1)-dimensional Shallow Water-like equation,” Journal of Computers & Mathematics with Applications, vol. 73, no. 2, pp. 246–252, 2017. View at: Publisher Site | Google Scholar | MathSciNet
8. C. Fu, C. N. Lu, and H. W. Yang, “Time-space fractional (2+1) dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions,” Advances in Difference Equations, vol. 2018, p. 56, 2018. View at: Publisher Site | Google Scholar | MathSciNet
9. H. W. Yang, M. Guo, and H. He, “Conservation laws of space-time fractional mZK equation for Rossby solitary waves with complete Coriolis force,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 20, no. 1, pp. 1–16, 2018. View at: Publisher Site | Google Scholar | MathSciNet
10. V. L. Dorofeyev and V. D. Larichev, “The exchange of fluid mass between quasi-geostrophic and ageostrophic motions during the reflection of Rossby waves from a coast. I. The case of an infinite rectilinear coast,” Dynamics of Atmospheres and Oceans, vol. 16, no. 3-4, pp. 305–329, 1992. View at: Publisher Site | Google Scholar
11. M. McAnally and W.-X. Ma, “An integrable generalization of the D-Kaup–Newell soliton hierarchy and its bi-Hamiltonian reduced hierarchy,” Applied Mathematics and Computation, vol. 323, pp. 220–227, 2018. View at: Publisher Site | Google Scholar | MathSciNet
12. H. Dong, Y. Zhang, and X. Zhang, “The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 36, pp. 354–365, 2016. View at: Publisher Site | Google Scholar | MathSciNet
13. X.-Y. Li, Q.-L. Zhao, and Y.-X. Li, “A new integrable symplectic map for 4-field Blaszak-Marciniak lattice equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 7, pp. 2324–2333, 2014. View at: Publisher Site | Google Scholar | MathSciNet
14. R. Zhou, “A Darboux transformation of the sl(2|1) super KDV hierarchy and a super lattice potential KDV Equation,” Physics Letters A, vol. 378, no. 26-27, pp. 1816–1819, 2014. View at: Publisher Site | Google Scholar | MathSciNet
15. X.-Y. Wen, “An integrable lattice hierarchy, associated integrable coupling, Darboux transformation and conservation laws,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5796–5805, 2012. View at: Publisher Site | Google Scholar | MathSciNet
16. B. Hu and T. Xia, “The binary nonlinearization of the super integrable system and its self-consistent sources,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 18, no. 3-4, pp. 285–292, 2017. View at: Publisher Site | Google Scholar | MathSciNet
17. X. Li and Q. Zhao, “A new integrable symplectic map by the binary nonlinearization to the super AKNS system,” Journal of Geometry and Physics, vol. 121, pp. 123–137, 2017. View at: Publisher Site | Google Scholar
18. X.-X. Xu and Y.-P. Sun, “Two symmetry constraints for a generalized Dirac integrable hierarchy,” Journal of Mathematical Analysis and Applications, vol. 458, no. 2, pp. 1073–1090, 2018. View at: Publisher Site | Google Scholar | MathSciNet
19. S. Yang and T. Xu, “Lie symmetry analysis for a parabolic Monge-Ampère equation in the optimal investment theory,” Journal of Computational and Applied Mathematics, vol. 346, pp. 483–489, 2019. View at: Publisher Site | Google Scholar | MathSciNet
20. G. Wang and K. Fakhar, “Lie symmetry analysis, nonlinear self-adjointness and conservation laws to an extended (2+1)-dimensional Zakharov-Kuznetsov-Burgers equation,” Computers & Fluids, vol. 119, pp. 143–148, 2015. View at: Publisher Site | Google Scholar
21. H. Wu, Y. Zeng, and T. Fan, “Complexitons of the modified KdV equation by Darboux transformation,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 501–510, 2008. View at: Publisher Site | Google Scholar | MathSciNet
22. J. Chen, Z. Ma, and Y. Hu, “Nonlocal symmetry, Darboux transformation and soliton-cnoidal wave interaction solution for the shallow water wave equation,” Journal of Mathematical Analysis and Applications, vol. 460, no. 2, pp. 987–1003, 2018. View at: Publisher Site | Google Scholar | MathSciNet
23. S. D. Zhu and J. F. Song, “Residual symmetries, the Bäcklund transformation and interaction solutions for (2+1)-dimensional generalized Broer-Kaup equations,” Applied Mathematics Letters, vol. 83, pp. 33–39, 2018. View at: Google Scholar
24. Q. L. Zhao, X. Y. Li, and F. S. Liu, “Two integrable lattice hierarchies and their respective Darboux transformations,” Applied Mathematics and Computation, vol. 219, no. 10, pp. 5693–5705, 2013. View at: Publisher Site | Google Scholar | MathSciNet
25. 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. View at: Publisher Site | Google Scholar | MathSciNet
26. W. Li, Y. Han, and G. Zhou, “Darboux transformation of a nonlinear evolution equation and its explicit solutions,” Acta Mathematica Scientia B, vol. 31, no. 4, pp. 1457–1464, 2011. View at: Publisher Site | Google Scholar | MathSciNet
27. X.-Y. Wen, “A new integrable lattice hierarchy associated with a discrete 3x3 matrix spectral problem: N-fold Darboux transformation and explicit solutions,” Reports on Mathematical Physics, vol. 71, no. 1, pp. 15–32, 2013. View at: Publisher Site | Google Scholar | MathSciNet
28. N. Zhang and T.-C. Xia, “A new negative discrete hierarchy and its N-fold darboux transformation,” Communications in Theoretical Physics, vol. 68, no. 6, pp. 687–692, 2017. View at: Publisher Site | Google Scholar
29. X.-Y. Wen, “N-soliton solutions and conservation laws of the modified Toda lattice equation,” Modern Physics Letters B, vol. 26, no. 6, Article ID 1150032, pp. 1–14, 2012. View at: Publisher Site | Google Scholar | MathSciNet
30. L. Liu, X.-Y. Wen, and D.-S. Wang, “A new lattice hierarchy: Hamiltonian structures, symplectic map and N-fold Darboux transformation,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 67, pp. 201–218, 2019. View at: Publisher Site | Google Scholar | MathSciNet
31. X. Geng, J. Shen, and B. Xue, “A new nonlinear wave equation: Darboux transformation and soliton solutions,” Wave Motion, vol. 79, pp. 44–56, 2018. View at: Publisher Site | Google Scholar | MathSciNet
32. H.-Q. Zhang and F. Chen, “Dark and antidark solitons for the defocusing coupled Sasa-SATsuma system by the Darboux transformation,” Applied Mathematics Letters, vol. 88, pp. 237–242, 2019. View at: Publisher Site | Google Scholar | MathSciNet
33. L.-Y. Ma, H.-Q. Zhao, S.-F. Shen, and W.-X. Ma, “Abundant exact solutions to the discrete complex mKdV equation by Darboux transformation,” Communications in Nonlinear Science and Numerical Simulation, vol. 68, pp. 31–40, 2019. View at: Publisher Site | Google Scholar | MathSciNet
34. S. S. Chen, B. Tian, L. Liu, Y. Q. Yuan, and C. R. Zhang, “Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system,” Chaos, Solitons & Fractals, vol. 118, pp. 337–346, 2019. View at: Publisher Site | Google Scholar
35. 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. View at: Publisher Site | Google Scholar | MathSciNet
36. W. X. Ma, “Symmetry constraint of MKdV equations by binary nonlinearization,” Physica A: Statistical and Theoretical Physics, vol. 219, no. 3-4, pp. 467–481, 1995. View at: Publisher Site | Google Scholar | MathSciNet
37. Y. Li and W. Ma, “Binary nonlinearization of AKNS spectral problem under higher-order symmetry constraints,” Chaos, Solitons & Fractals, vol. 11, no. 5, pp. 697–710, 2000. View at: Publisher Site | Google Scholar | MathSciNet
38. W. X. Ma, B. Fuchssteiner, and W. Oevel, “A 3 x 3 matrix spectral problem for AKNS hierarchy and its binary nonlinearization,” Physica A. Statistical and Theoretical Physics, vol. 233, no. 1-2, pp. 331–354, 1996. View at: Publisher Site | Google Scholar | MathSciNet

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