On a Toda lattice hierarchy: Lax pair, integrable symplectic map and algebraic–geometric solution

A Toda lattice hierarchy is studied by introducing a new spectral problem which is a discrete counterpart of the generalized Kaup–Newell spectral problem. Based on the Lenard recursion equation, Lax pair of the hierarchy is given. Further, the discrete spectral problem is nonlinearized into an integrable symplectic map. As a result, an algebraic–geometric solution in Riemann theta function of the hierarchy is obtained. Besides, two equations, the Volterra lattice and a (2[Formula: see text]+[Formula: see text]1)-dimensional Burgers equation with a discrete variable, yielded from the hierarchy are also solved.

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Super Riemann theta function periodic wave solutions and rational characteristics for a supersymmetric KdV-Burgers equation

Theoretical and Mathematical Physics ◽

10.1007/s11232-012-0031-8 ◽

2012 ◽

Vol 170 (3) ◽

pp. 287-314 ◽

Cited By ~ 13

Author(s):

Shou-fu Tian ◽

Hong-qing Zhang

Keyword(s):

Theta Function ◽

Burgers Equation ◽

Periodic Wave ◽

Riemann Theta Function ◽

Periodic Wave Solutions ◽

Wave Solutions

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A Toda lattice hierarchy with variable coefficients and its multi-wave solutions

Thermal Science ◽

10.2298/tsci1405563z ◽

2014 ◽

Vol 18 (5) ◽

pp. 1563-1566 ◽

Cited By ~ 3

Author(s):

Sheng Zhang ◽

Wang Di

Keyword(s):

Spectral Problem ◽

Toda Lattice ◽

Variable Coefficients ◽

Curvature Equation ◽

Wave Solutions ◽

Zero Curvature ◽

N Wave ◽

Special Case ◽

Toda Lattice Hierarchy

Starting from the Toda spectral problem, a new Toda lattice hierarchy of isospectral equations with variable coefficients is constructed through the discrete zero curvature equation. In order to solve one special case of the derived Toda lattice hierarchy, a series of appropriate transformations are utilized. As a result, a new uniform formula of N-wave solutions is obtained.

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Decomposing the Toda hierarchy by an implicit symmetry constraint

Modern Physics Letters B ◽

10.1142/s0217984919500283 ◽

2019 ◽

Vol 33 (03) ◽

pp. 1950028

Author(s):

Xi-Xiang Xu ◽

Min Guo ◽

Ning Zhang

Keyword(s):

Hamiltonian System ◽

Toda Lattice ◽

Lattice Equation ◽

Symmetry Constraint ◽

Completely Integrable ◽

Toda Hierarchy ◽

Finite Dimensional ◽

Symplectic Map ◽

Toda Lattice Hierarchy

An implicit symmetry constraint of the famous Toda lattice hierarchy is presented. Using this symmetry constraint, every lattice equation in the Toda hierarchy is decomposed by an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system.

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Generalized fermionic discrete Toda hierarchy

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022604311027 ◽

2004 ◽

Vol 2004 (1) ◽

pp. 113-153 ◽

Cited By ~ 5

Author(s):

V. V. Gribanov ◽

V. G. Kadyshevsky ◽

A. S. Sorin

Keyword(s):

Boundary Conditions ◽

Periodic Boundary ◽

Hamiltonian Structure ◽

Toda Lattice ◽

Monodromy Matrix ◽

Periodic Boundary Conditions ◽

Lax Pair ◽

Spectral Curves ◽

Matrix Description ◽

Toda Lattice Hierarchy

We describe bi-Hamiltonian structure and Lax-pair formulation with the spectral parameter of the generalized fermionic Toda lattice hierarchy as well as its bosonic and fermionic symmetries for different (including periodic) boundary conditions. Its two reductionsN=4andN=2supersymmetric Toda lattice hierarchiesin different (including canonical) bases are investigated. Itsr-matrix description, monodromy matrix, and spectral curves are discussed.

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ON THE INTEGRATION OF A HIRERARCHY FOR THE PEREODIC TODA LATTICE WITH A FREE TERM

Central Asian Problems of Modern Science and Education ◽

10.51348/campse0024 ◽

2021 ◽

pp. 101-110

Keyword(s):

Spectral Problem ◽

Spectral Data ◽

Time Evolution ◽

Toda Lattice ◽

Inverse Spectral Problem ◽

Complete Integrability ◽

Free Term ◽

Periodic Functions ◽

Periodic Coefficients ◽

Toda Lattice Hierarchy

In this article, we have explored the Toda lattice hierarchy in the class of periodic functions with a free term. We have given an effective method of constructing of the periodic Toda lattice hierarchy with a free term. We have discussed the complete integrability of the constructed systems that is based on the inverse spectral problem of an associated discrete Hill`s equation with periodic coeﬃcients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy.

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Mukherjee–Choudhury–Chowdhury spectral problem and the semi-discrete integrable system

International Journal of Modern Physics B ◽

10.1142/s0217979216400270 ◽

2016 ◽

Vol 30 (28n29) ◽

pp. 1640027

Author(s):

Xi-Xiang Xu ◽

Ye-Peng Sun

Keyword(s):

Spectral Problem ◽

Integrable System ◽

Bäcklund Transformation ◽

Proper Time ◽

Lax Pair ◽

Hamiltonian Form ◽

Backlund Transformation ◽

Discrete Integrable System ◽

Finite Dimensional ◽

Symplectic Map

Starting from the Mukherjee–Choudhury–Chowdhury spectral problem, we derive a semi-discrete integrable system by a proper time spectral problem. A Bäcklund transformation of Darboux type of this system is established with the help of gauge transformation of the Lax pairs. By means of the obtained Bäcklund transformation, an exact solution is given. Moreover, Hamiltonian form of this system is constructed. Further, through a constraint of potentials and eigenfunctions, the Lax pair and the adjoint Lax pair of the obtained semi-discrete integrable system are nonlinearized as an integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system in the Liouville sense. Finally, the involutive representation of solution of the obtained semi-discrete integrable system is presented.

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On (2+1)-Dimensional Non-isospectral Toda Lattice Hierarchy and Integrable Coupling System

Communications in Theoretical Physics ◽

10.1088/0253-6102/49/3/05 ◽

2008 ◽

Vol 49 (3) ◽

pp. 549-554 ◽

Cited By ~ 1

Author(s):

Yu Fa-Jun

Keyword(s):

Toda Lattice ◽

Coupling System ◽

Integrable Coupling ◽

Toda Lattice Hierarchy

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REDUCTION OF TODA LATTICE HIERARCHY TO GENERALIZED KdV HIERARCHIES AND THE TWO-MATRIX MODEL

International Journal of Modern Physics A ◽

10.1142/s0217751x95001212 ◽

1995 ◽

Vol 10 (17) ◽

pp. 2537-2577 ◽

Cited By ~ 12

Author(s):

H. ARATYN ◽

E. NISSIMOV ◽

S. PACHEVA ◽

A.H. ZIMERMAN

Keyword(s):

Poisson Bracket ◽

Hamiltonian Structure ◽

Toda Lattice ◽

Free Field ◽

String Model ◽

Matrix Formulation ◽

Hamiltonian Action ◽

Kdv Hierarchy ◽

Associated Matrix ◽

Toda Lattice Hierarchy

Toda lattice hierarchy and the associated matrix formulation of the 2M-boson KP hierarchies provide a framework for the Drinfeld-Sokolov reduction scheme realized through Hamiltonian action within the second KP Poisson bracket. By working with free currents, which Abelianize the second KP Hamiltonian structure, we are able to obtain a unified formalism for the reduced SL (M+1, M−k) KdV hierarchies interpolating between the ordinary KP and KdV hierarchies. The corresponding Lax operators are given as superdeterminants of graded SL (M+1, M−k) matrices in the diagonal gauge and we describe their bracket structure and field content. In particular, we provide explicit free field representations of the associated W(M, M−k) Poisson bracket algebras generalizing the familiar nonlinear WM+1 algebra. Discrete Bäcklund transformations for SL (M+1, M−k) KdV are generated naturally from lattice translations in the underlying Toda-like hierarchy. As an application we demonstrate the equivalence of the two-matrix string model to the SL (M+1, 1) KdV hierarchy.

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