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.

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.

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 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.

CONTINUUM REDUCTION OF VIRASORO-CONSTRAINED LATTICE HIERARCHIES

1991 ◽  
Vol 06 (28) ◽  
pp. 2601-2612 ◽  
Author(s):  
A. M. SEMIKHATOV
Keyword(s):  
Continuum Limit ◽  
Toda Lattice ◽  
Integrable Hierarchies ◽  
Scaling Limit ◽  
Explicit Construction ◽  
Kp Hierarchy ◽  
Highest Weight ◽  
Virasoro Constraints ◽  
Toda Lattice Hierarchy ◽  
The Continuum

Integrable hierarchies with Virasoro constraints have been observed to describe matrix models. I suggest to define general Virasoro-constrained integrable hierarchies by imposing Virasora-highest-weight conditions on the dressing operators. This simplifies the study of the Virasoro constraints and allows an explicit construction of a scaling which implements the continuum limit of discrete (lattice) hierarchies. Applied to the Toda lattice hierarchy subjected to the Virasoro constraints, this scaling leads to the Virasoro-constrained KP hierarchy. Therefore, in particular, the KP hierarchy is shown to arise as the scaling limit of a matrix model.

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

1995 ◽  
Vol 10 (17) ◽  
pp. 2537-2577 ◽  
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.

The generalized additional symmetries of the two-Toda lattice hierarchy

2013 ◽  
Vol 54 (2) ◽  
pp. 023513 ◽  
Author(s):  
Jipeng Cheng ◽  
Ye Tian ◽  
Zhaowen Yan ◽  
Jingsong He
Keyword(s):  
Toda Lattice ◽  
Additional Symmetries ◽  
Toda Lattice Hierarchy

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

2021 ◽  
Author(s):  
Zhiguo Xu
Keyword(s):  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Integrable Hierarchy ◽  
Integrable Lattice ◽  
Pass Through ◽  
Matrix Spectral Problem ◽  
Relativistic Toda Lattice ◽  
Toda Lattice Hierarchy

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

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