Decomposing the Toda hierarchy by an implicit symmetry constraint

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.

scholarly journals Bargmann Type Systems for the Generalization of Toda Lattices

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.

scholarly journals INTEGRABLE HIERARCHIES AND DISPERSIONLESS LIMIT

1995 ◽  
Vol 07 (05) ◽  
pp. 743-808 ◽  
Author(s):  
KANEHISA TAKASAKI ◽  
TAKASHI TAKEBE
Keyword(s):  
Canonical Transformation ◽  
Toda Lattice ◽  
Integrable Hierarchies ◽  
General Solutions ◽  
Similar Construction ◽  
Toda Hierarchy ◽  
Technical Details ◽  
Dispersionless Limit ◽  
Toda Lattice Hierarchy ◽  
Twistor Construction

Analogues of the KP and the Toda lattice hierarchy called dispersionless KP and Toda hierarchy are studied. Dressing operations in the dispersionless hierarchies are introduced as a canonical transformation, quantization of which is dressing operators of the ordinary KP and Toda hierarchy. An alternative construction of general solutions of the ordinary KP and Toda hierarchy is given as twistor construction which is quantization of the similar construction of solutions of dispersionless hierarchies. These results as well as those obtained in previous papers are presented with proofs and necessary technical details.

THE NONLINEARIZATION OF A (2+1)-DIMENSIONAL SOLITON EQUATION

2008 ◽  
Vol 22 (32) ◽  
pp. 3179-3194
Author(s):  
QIANG LIU ◽  
DIAN-LOU DU
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Hamiltonian System ◽  
Eigenvalue Problem ◽  
Hamiltonian Systems ◽  
Soliton Equation ◽  
Completely Integrable ◽  
Finite Dimensional ◽  
Dimensional Soliton

Based on a 2 × 2 eigenvalue problem, a new (2+1)-dimensional soliton equation is proposed. Moreover, we obtain a finite-dimensional Hamiltonian system. Then we verify it is completely integrable in the Liouville sense. In the end, we introduce a set of Hk polynomial integrable, by which we can separate the solition equation into three compantiable Hamiltonian systems of ordinary differential equation.

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

2018 ◽  
Vol 32 (28) ◽  
pp. 1850344
Author(s):  
Xiao Yang ◽  
Dianlou Du
Keyword(s):  
Spectral Problem ◽  
Theta Function ◽  
Toda Lattice ◽  
Burgers Equation ◽  
Lax Pair ◽  
Discrete Variable ◽  
Riemann Theta Function ◽  
Symplectic Map ◽  
Discrete Counterpart ◽  
Toda Lattice Hierarchy

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.

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

Open Physics ◽  
2013 ◽  
Vol 11 (1) ◽  
Author(s):  
Anindya Choudhury
Keyword(s):  
Generating Function ◽  
Toda Lattice ◽  
Bäcklund Transformation ◽  
Hamiltonian Formalism ◽  
Bäcklund Transformations ◽  
Lattice Equation ◽  
Toda Hierarchy ◽  
Relativistic Toda Lattice ◽  
Toda Lattice Equation ◽  
Parameter Dependent

AbstractIn this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.

scholarly journals WHITHAM-TODA HIERARCHY AND N=2 SUPERSYMMETRIC YANG-MILLS THEORY

1996 ◽  
Vol 11 (02) ◽  
pp. 157-168 ◽  
Author(s):  
TOSHIO NAKATSU ◽  
KANEHISA TAKASAKI
Keyword(s):  
Moduli Space ◽  
Toda Lattice ◽  
Homogeneous Solution ◽  
Hyperelliptic Curves ◽  
Low Energy ◽  
Integrable Hierarchy ◽  
Quasiperiodic Solution ◽  
Yang Mills ◽  
Toda Hierarchy ◽  
Toda Lattice Hierarchy

The exact solution of N=2 supersymmetric SU(N) Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy (Whitham-Toda hierarchy) describes modulation of a quasiperiodic solution of the (generalized) Toda lattice hierarchy associated with the hyperelliptic curves over the quantum moduli space. The relation between the holomorphic pre-potential of the low energy effective action and the τ-function of the (generalized) Toda lattice hierarchy is also clarified.

scholarly journals FAY-LIKE IDENTITIES OF THE TODA LATTICE HIERARCHY AND ITS DISPERSIONLESS LIMIT

2006 ◽  
Vol 18 (10) ◽  
pp. 1055-1073 ◽  
Author(s):  
LEE-PENG TEO
Keyword(s):  
Toda Lattice ◽  
Tau Function ◽  
Toda Hierarchy ◽  
Dispersionless Limit ◽  
Toda Lattice Hierarchy ◽  
Hirota Equations

In this paper, we derive the Fay-like identities of tau function for the Toda lattice hierarchy from the bilinear identity. We prove that the Fay-like identities are equivalent to the hierarchy. We also show that the dispersionless limit of the Fay-like identities are the dispersionless Hirota equations of the dispersionless Toda hierarchy.

VARIABLE SEPARATION AND ALGEBRAIC-GEOMETRIC SOLUTIONS OF MODIFIED TODA LATTICE EQUATION

2010 ◽  
Vol 24 (19) ◽  
pp. 2041-2055
Author(s):  
LIN LUO ◽  
ENGUI FAN
Keyword(s):  
Toda Lattice ◽  
Theta Functions ◽  
Variable Separation ◽  
Lattice Equation ◽  
Riemann Theta Functions ◽  
Jacobi Inversion ◽  
Toda Lattice Equation ◽  
Hyper Elliptic Curve ◽  
Geometric Solutions ◽  
Toda Lattice Hierarchy

In this paper, we derive a modified Toda lattice hierarchy by resorting to the Lenard operator pairs. The hyper-elliptic curve and Abel–Jacobi coordinates are then introduced to linearize the associated flow, from which some algebraic-geometric solutions of the modified Toda lattice are explicitly constructed in terms of Riemann theta functions by using Jacobi inversion technique.

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