A GENERALIZATION OF TODA LATTICES AND THEIR BI-HAMILTONIAN STRUCTURES

A hierarchy of new nonlinear differential-difference equations associated with fourth-order discrete spectral problem is proposed, in which a typical member is a generalization of the Toda lattice equation. The bi-Hamiltonian structures for this hierarchy are obtained with the help of trace identity.

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A HIERARCHY OF LIOUVILLE INTEGRABLE LATTICE EQUATION ASSOCIATED WITH A THREE-BY-THREE DISCRETE SPECTRAL PROBLEM AND ITS INFINITELY MANY CONSERVATION LAWS

International Journal of Modern Physics B ◽

10.1142/s0217979211100321 ◽

2011 ◽

Vol 25 (18) ◽

pp. 2481-2492

Author(s):

YU-QING LI ◽

XI-XIANG XU

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Trace Identity ◽

Lattice Equation ◽

Hamiltonian Structures ◽

Discrete Soliton ◽

Soliton Equations ◽

Integrable Lattice ◽

Matrix Spectral Problem ◽

Liouville Integrable

A discrete three-by-three matrix spectral problem is put forward and the corresponding discrete soliton equations are deduced. By means of the trace identity the Hamiltonian structures of the resulting equations are constructed, and furthermore, infinitely many conservation laws of the corresponding lattice system are obtained by a direct way.

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DARBOUX TRANSFORMATION OF THE MODIFIED TODA LATTICE EQUATION

Modern Physics Letters B ◽

10.1142/s0217984906011025 ◽

2006 ◽

Vol 20 (11) ◽

pp. 641-648 ◽

Cited By ~ 15

Author(s):

XI-XIANG XU ◽

HONG-XIANG YANG ◽

YE-PENG SUN

Keyword(s):

Spectral Problem ◽

Soliton Solution ◽

Darboux Transformation ◽

Toda Lattice ◽

Lattice Equation ◽

Matrix Spectral Problem ◽

Toda Lattice Equation

A modified Toda lattice equation associated with a properly discrete matrix spectral problem is introduced. Darboux transformation for the resulting lattice equation is constructed. As an application, the soliton solution for the Toda lattice equation is explicitly given.

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New hierarchy of integrable nonlinear differential-difference equations

Modern Physics Letters B ◽

10.1142/s0217984915501900 ◽

2015 ◽

Vol 29 (31) ◽

pp. 1550190

Author(s):

Xianguo Geng ◽

Liang Guan ◽

Bo Xue

Keyword(s):

Conservation Laws ◽

Spectral Problem ◽

Spectral Parameter ◽

Difference Equations ◽

Trace Identity ◽

Hamiltonian Structures ◽

Zero Curvature ◽

Matrix Spectral Problem

A hierarchy of integrable nonlinear differential-difference equations associated with a discrete [Formula: see text] matrix spectral problem is proposed based on the discrete zero-curvature equations. Then, Hamiltonian structures for this hierarchy are constructed with the aid of the trace identity. Infinitely many conservation laws of the hierarchy are derived by means of spectral parameter expansions.

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Bargmann Type Systems for the Generalization of Toda Lattices

Journal of Applied Mathematics ◽

10.1155/2014/287529 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Fang Li ◽

Liping Lu

Keyword(s):

Hamiltonian Systems ◽

Toda Lattice ◽

Type Systems ◽

Integrals Of Motion ◽

Lattice Equation ◽

Finite Dimensional ◽

Symplectic Map ◽

Representation Of Solutions ◽

Toda Lattice Equation ◽

Toda Lattices

Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete hierarchy of a generalization of the Toda lattice equation is proposed, which leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Finally, the representation of solutions for a lattice equation in the discrete hierarchy is obtained.

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