On periodic motions of a two dimensional Toda type chain (original) (raw)

On the periodic Toda lattice with a self-consistent source

Communications in Nonlinear Science and Numerical Simulation

This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice with self-consistent source. The effective method of solution of the inverse spectral problem for the discrete Hill’s equation is presented.

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.

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.

Near-integrability of periodic FPU-chains

Physica A: Statistical Mechanics and its Applications, 2000

The FPU-chain with periodic boundary conditions is studied by Birkho-Gustavson normalisation. In the cases of up to 6 particles and for ÿ-chains with an odd number of particles the normal forms are integrable, which permits us to apply KAM-theory. This leads to the presence of many invariant tori on which the motion is quasi-periodic. Thus we explain the recurrence phenomena and the small size of chaos observed in experiments. Furthermore, we ÿnd a certain clustering of modes.

The classical Toda lattice with long range interaction

Physics Letters A, 2003

In this Letter, I introduce a generalization of Toda lattice model, in which each site can interact with all other sites with different coupling constants. To show solvability of this model, by introducing new type of quantities (which are referred to as anti-Grassmanian numbers), the equations of motion are transformed to a Lax equation and constants of motion are derived. At last, by finding a proper r matrix involutivity of these constants are shown.  2002 Elsevier Science B.V. All rights reserved. 0375-9601/02/$ -see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 -9 6 0 1 ( 0 2 ) 0 0 1 7 0 -6

Traveling waves in systems of oscillators on 2D-lattices

Journal of Mathematical Sciences, 2011

A system of differential equations that describes the dynamics of an infinite system of linearly coupled nonlinear oscillators on a 2D-lattice is considered. The exponential estimate of the solution and some results on the existence of periodic and solitary traveling waves are obtained.

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.