Generation of discrete integrable systems and some algebrogeometric properties of related discrete lattice equations
Authors
Yufeng Zhang
 College of Mathematics, China University of Mining and Technology, Xuzhou 221116, P. R. China.
XiaoJun Yang
 School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, P. R. China.
 State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining and Technology, 221116, P. R. China.
Abstract
With the help of infinitedimensional 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 algebrogeometric solutions. Besides, a reduced spectral problem is given to find an integrable
discrete hierarchy obtained via Rmatrix 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, XiaoJun Yang, Generation of discrete integrable systems and some algebrogeometric properties of related discrete lattice equations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 61266141
AMA Style
Zhang Yufeng, Yang XiaoJun, Generation of discrete integrable systems and some algebrogeometric properties of related discrete lattice equations. J. Nonlinear Sci. Appl. (2016); 9(12):61266141
Chicago/Turabian Style
Zhang, Yufeng, Yang, XiaoJun. "Generation of discrete integrable systems and some algebrogeometric properties of related discrete lattice equations." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 61266141
Keywords
 Spectral problem
 algebrogeometric solution
 Rmatrix
 Hamiltonian structure.
MSC
 35P30
 35Q51
 37K05
 37K10
 37K15
 37K30
References

[1]
M. Blaszak, K. Marciniak, Rmatrix approach to lattice integrable systems, J. Math. Phys., 35 (1994), 46614682

[2]
M. Blaszak, A. Szum, Lie algebraic approach to the construction of (2 + 1)dimensional latticefield and field integrable Hamiltonian equations, J. Math. Phys., 42 (2001), 225259

[3]
M. Blaszak, A. Szum, A. Prykarpatsky, Central extension approach to integrable field and latticefield systems in (2 + 1)dimensions, Proceedings of the XXX Symposium on Mathematical Physics, ToruĊ, (1998), Rep. Math. Phys., 44 (1999), 3744

[4]
E.G. Fan, Z.H. Yang, A lattice hierarchy with a free function and its reductions to the AblowitzLadik and Volterra hierarchies, Internat. J. Theoret. Phys., 48 (2009), 19

[5]
X.G. Geng, H. H. Dai, Quasiperiodic solutions for some 2+1dimensional discrete models, Phys. A, 319 (2003), 270294

[6]
X.G. Geng, H. H. Dai, C.W. Cao, Algebrogeometric constructions of the discrete AblowitzLadik flows and applications, J. Math. Phys., 44 (2003), 45737588

[7]
Y. C. Hon, E. G. Fan, An algebrogeometric 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 semidirect sums of Lie algebras, J. Phys. A, 40 (2007), 1505515069

[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), 24002418

[10]
A. Pickering, Z.N. Zhu, New integrable lattice hierarchies, Phys. Lett. A, 349 (2006), 439445

[11]
A. Pickering, Z.N. Zhu, DarbouxBäcklund transformation and explicit solutions to a hybrid lattice of the relativistic Toda lattice and the modified Toda lattice, Phys. Lett. A, 378 (2014), 15101513

[12]
Z.J. Qiao, A hierarchy of nonlinear evolution equations and finitedimensional involutive systems, J. Math. Phys., 35 (1994), 29712992

[13]
Z.J. Qiao, Generalized rmatrix structure and algebrogeometric solution for integrable system, Rev. Math. Phys., 13 (2001), 545586

[14]
Z.J. Qiao, The CamassaHolm hierarchy, Ndimensional integrable systems, and algebrogeometric solution on a symplectic submanifold, Comm. Math. Phys., 239 (2003), 309341

[15]
M. Toda, Theory of nonlinear lattices, Translated from the Japanese by the author, Springer Series in SolidState Sciences, SpringerVerlag, BerlinNew York (1981)

[16]
G. Z. Tu, A trace identity and its applications to the theory of discrete integrable systems, J. Phys. A, 23 (1990), 39033922

[17]
Y.F. Zhang, B.L. Feng, W.J. Rui, X.Z. Zhang, Algebrogeometric solutions with characteristics of a nonlinear partial differential equation with threepotential functions, Commun. Theor. Phys. (Beijing), 64 (2015), 8189

[18]
Y.F. Zhang, W.J. Rui, A few continuous and discrete dynamical systems, Rep. Math. Phys., 78 (2016), 1932

[19]
R.G. Zhou, The finiteband solution of the JaulentMiodek equation, J. Math. Phys., 38 (1997), 25352546

[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), 77357742

[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 Todatype lattice soliton equations, J. Phys. A, 32 (1999), 41714182