A note on the superintegrability of the Toda lattice (original) (raw)
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On the Integrable Generalization of the 1D Toda Lattice
2010
A generalized Toda Lattice equation is considered. The associated linear problem (Lax representation) is found. For simple case N = 3 the τ -function Hirota form is presented that allows to construct an exast solutions of the equations of the 1DGTL. The corresponding hierarchy and its relations with the nonlinear Schrodinger equation and Hersenberg ferromagnetic equation are discussed.
On some integrable systems related to the Toda lattice
Journal of Physics A: Mathematical and General, 1997
A new integrable lattice system is introduced, and its integrable discretizations are obtained. A Bäcklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established.
New integrable systems related to the relativistic Toda lattice
Journal of Physics A: Mathematical and General, 1997
New integrable lattice systems are introduced, their different integrable discretization are obtained. Bäcklund transformations between these new systems and the relativistic Toda lattice (in the both continuous and discrete time formulations) are established.
The finite non-periodic toda lattice: A geometric and topological viewpoint
2008
In 1967, Japanese physicist Morikazu Toda published the seminal papers [78] and [79], exhibiting soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the decades that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics that sets it apart from the others. These are known collectively as the Toda lattice. This survey describes and compares several versions of the finite non-periodic Toda lattice from the perspective of their geometry and topology. Contents 1. Outline of the paper 2 2. Finite non-periodic real Toda lattice 3 2.1. Symmetric form 4 2.2. Hessenberg form 6 2.3. Extended real tridiagonal symmetric form 7 2.4. Extended real tridiagonal Hessenberg form 9 2.5. Full Symmetric real Toda lattice 3. Complex Toda lattices 3.1. The moment map 3.2. Complex tridiagonal Hessenberg form 3.3. The full Kostant-Toda lattice 3.4. Nongeneric flows in the full Kostant-Toda lattice 4. Other Extensions of the Toda Lattice 4.1. Isospectral deformation of a general matrix 4.2. Gradient formulation of Toda flows 5. Connections with the KP equation 5.1. The τ-functions for the symmetric Toda lattice hierarchy 5.2. The KP equation and the τ-function 5.3. Grassmannian Gr(k, n) 6. The Toda lattice and integral cohomology of real flag manifolds 6.1. The moment polytope and Weyl group action 6.2. Integral cohomology of G/B 6.3. Blow-ups of the indefinite Toda lattice on G and the cohomology of G/B References 1 Partially supported by NSF grant DMS0404931.
Fifty years of the finite nonperiodic Toda lattice: a geometric and topological viewpoint
Journal of Physics A: Mathematical and Theoretical, 2018
In 1967, Japanese physicist Morikazu Toda published a pair of seminal papers in the Journal of the Physical Society of Japan that exhibited soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the fifty years that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics. These are known collectively as the Toda lattice. This survey recounts and compares the various versions of the finite nonperiodic Toda lattice from the perspective of their geometry and topology. In particular, we highlight the polytope structure of the solution spaces as viewed through the moment map, and we explain the connection between the real indefinite Toda flows and the integral cohomology of real flag varieties.
Change of the Time for the Toda Lattice
Journal of Nonlinear Mathematical Physics, 2001
For the Toda lattice we consider properties of the canonical transformations of the extended phase space, which preserve integrability. At the special values of integrals of motion the integral trajectories, separated variables and the action variables are invariant under change of the time. On the other hand, mapping of the time induces shift of the generating function of the Bäcklund transformation.
Journal of Physics A: Mathematical and General, 2000
We consider compositions of the transformations of the time variable and canonical transformations of the other coordinates, which map completely integrable system into other completely integrable system. Change of the time gives rise to transformations of the integrals of motion and the Lax pairs, transformations of the corresponding spectral curves and R-matrices. As an example, we consider canonical transformations of the extended phase space for the Toda lattices and the Stäckel systems.
On the periodic Toda lattice hierarchy with an integral source
Communications in Nonlinear Science and Numerical Simulation, 2017
This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice hierarchy with an integral type source. The effective method is presented of constructing the periodic Toda lattice hierarchy with an integral source.