Generation of discrete integrable systems and some algebro-geometric properties of related discrete lattice equations
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Authors
Yufeng Zhang
- College of Mathematics, China University of Mining and Technology, Xuzhou 221116, P. R. China.
Xiao-Jun Yang
- School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, P. R. China.
- State Key Laboratory for Geo-Mechanics and Deep Underground Engineering, China University of Mining and Technology, 221116, P. R. China.
Abstract
With the help of infinite-dimensional Lie algebras and the Tu scheme, we address a discrete integrable
hierarchy to reduce the generalized relativistic Toda lattice (GRTL) system containing the relativistic Toda
lattice equation and its generalized lattice equation. Meanwhile, the Riemann theta functions are utilized
to present its algebro-geometric solutions. Besides, a reduced spectral problem is given to find an integrable
discrete hierarchy obtained via R-matrix theory, which can be reduced to the Toda lattice equation and a
generalized Toda lattice (GTL) system. The Lax pair and the infinite conservation laws of the GTL system
are also derived. Finally, the Hamiltonian structure of the GTL system is generated by the Poisson tensor.
Share and Cite
ISRP Style
Yufeng Zhang, Xiao-Jun Yang, Generation of discrete integrable systems and some algebro-geometric properties of related discrete lattice equations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 6126--6141
AMA Style
Zhang Yufeng, Yang Xiao-Jun, Generation of discrete integrable systems and some algebro-geometric properties of related discrete lattice equations. J. Nonlinear Sci. Appl. (2016); 9(12):6126--6141
Chicago/Turabian Style
Zhang, Yufeng, Yang, Xiao-Jun. "Generation of discrete integrable systems and some algebro-geometric properties of related discrete lattice equations." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 6126--6141
Keywords
- Spectral problem
- algebro-geometric solution
- R-matrix
- Hamiltonian structure.
MSC
- 35P30
- 35Q51
- 37K05
- 37K10
- 37K15
- 37K30
References
-
[1]
M. Blaszak, K. Marciniak, R-matrix approach to lattice integrable systems, J. Math. Phys., 35 (1994), 4661--4682
-
[2]
M. Blaszak, A. Szum, Lie algebraic approach to the construction of (2 + 1)-dimensional lattice-field and field integrable Hamiltonian equations, J. Math. Phys., 42 (2001), 225--259
-
[3]
M. Blaszak, A. Szum, A. Prykarpatsky, Central extension approach to integrable field and lattice-field systems in (2 + 1)-dimensions, Proceedings of the XXX Symposium on Mathematical Physics, ToruĊ, (1998), Rep. Math. Phys., 44 (1999), 37--44
-
[4]
E.-G. Fan, Z.-H. Yang, A lattice hierarchy with a free function and its reductions to the Ablowitz-Ladik and Volterra hierarchies, Internat. J. Theoret. Phys., 48 (2009), 1--9
-
[5]
X.-G. Geng, H. H. Dai, Quasi-periodic solutions for some 2+1-dimensional discrete models, Phys. A, 319 (2003), 270--294
-
[6]
X.-G. Geng, H. H. Dai, C.-W. Cao, Algebro-geometric constructions of the discrete Ablowitz-Ladik flows and applications, J. Math. Phys., 44 (2003), 4573--7588
-
[7]
Y. C. Hon, E. G. Fan, An algebro-geometric solution for a Hamiltonian system with application to dispersive long wave equation, J. Math. Phys., 46 (2005), 21 pages
-
[8]
W.-X. Ma, A discrete variational identity on semi-direct sums of Lie algebras, J. Phys. A, 40 (2007), 15055--15069
-
[9]
W.-X. Ma, B. Fuchssteiner, Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations, J. Math. Phys., 40 (1990), 2400--2418
-
[10]
A. Pickering, Z.-N. Zhu, New integrable lattice hierarchies, Phys. Lett. A, 349 (2006), 439--445
-
[11]
A. Pickering, Z.-N. Zhu, Darboux-Bäcklund transformation and explicit solutions to a hybrid lattice of the relativistic Toda lattice and the modified Toda lattice, Phys. Lett. A, 378 (2014), 1510--1513
-
[12]
Z.-J. Qiao, A hierarchy of nonlinear evolution equations and finite-dimensional involutive systems, J. Math. Phys., 35 (1994), 2971--2992
-
[13]
Z.-J. Qiao, Generalized r-matrix structure and algebro-geometric solution for integrable system, Rev. Math. Phys., 13 (2001), 545--586
-
[14]
Z.-J. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys., 239 (2003), 309--341
-
[15]
M. Toda, Theory of nonlinear lattices, Translated from the Japanese by the author, Springer Series in Solid-State Sciences, Springer-Verlag, Berlin-New York (1981)
-
[16]
G. Z. Tu, A trace identity and its applications to the theory of discrete integrable systems, J. Phys. A, 23 (1990), 3903--3922
-
[17]
Y.-F. Zhang, B.-L. Feng, W.-J. Rui, X.-Z. Zhang, Algebro-geometric solutions with characteristics of a nonlinear partial differential equation with three-potential functions, Commun. Theor. Phys. (Beijing), 64 (2015), 81--89
-
[18]
Y.-F. Zhang, W.-J. Rui, A few continuous and discrete dynamical systems, Rep. Math. Phys., 78 (2016), 19--32
-
[19]
R.-G. Zhou, The finite-band solution of the Jaulent-Miodek equation, J. Math. Phys., 38 (1997), 2535--2546
-
[20]
R.-G. Zhou, Q.-Y. Jiang, A Darboux transformation and an exact solution for the relativistic Toda lattice equation, J. Phys. A, 38 (2005), 7735--7742
-
[21]
Z.-N. Zhu, Discrete zero curvature representations and infinitely many conservation laws for several 2+1 dimensional lattice hierarchies, ArXiv, 2003 (2003), 18 pages
-
[22]
Z.-N. Zhu, H.-C. Huang, Integrable discretizations for Toda-type lattice soliton equations, J. Phys. A, 32 (1999), 4171--4182