In this paper we compute the Birkhoff normal form of the periodic Toda lattice up to order four. As an application, we verify that Kolmogorov's nondegeneracy condition in the KAM theorem holds almost everywhere in phase space.

2006-09-18math-phmath.MPnlin.SI

Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions

Symmetries of the periodic Toda lattice are expresssed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Jacobi matrices. Using these symmetries, the phase space of the lattice with Dirichlet boundary conditions is embedded into the phase space of a higher-dimensional periodic lattice. As an application, we obtain a Birkhoff normal form and a KAM theorem for the lattice with Dirichlet boundary conditions.

2013-12-19math-phmath.MPnlin.SI

Factorization methods for Noncommutative KP and Toda hierarchy

We show that the solution space of the noncommutative KP hierarchy is the same as that of the commutative KP hierarchy owing to the Birkhoff decomposition of groups over the noncommutative algebra. The noncommutative Toda hierarchy is introduced. We derive the bilinear identities for the Baker--Akhiezer functions and calculate the $N$-soliton solutions of the noncommutative Toda hierarchy.

2005-04-11nlin.SI

Normal forms for Lie symmetric cotangent bundle systems with free and proper actions

We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space suitable for the study of dynamics near relative equilibria, in particular for the Birkhoff-Poincare normal form method. For a general symmetry group, we observe that for the calculation of the truncated normal forms, one does not need an explicit coordinate transformation but only its higher derivatives at the relative equilibrium. We outline an iterative schem…

2013-11-29math-phmath.DGmath.DS

On the Darboux-Nijenhuis Variables for the Open Toda Lattice

We discuss two known constructions proposed by Moser and by Sklyanin of the Darboux-Nijenhuis coordinates for the open Toda lattice.

2006-12-30math-phmath.MPnlin.SI

Semi-invariants and Integrals of the Full Symmetric sl(n) Toda Lattice

We consider the full symmetric version of the Lax operator of the Toda lattice which is known as the full symmetric Toda lattice. The phase space of this system is the generic orbit of the coadjoint action of the Borel subgroup B^+(n) of SL(n,R). This system is integrable. We propose a new method of constructing semi-invariants and integrals of the full symmetric Toda lattice. Using only the Toda equations for the Lax eigenvector matrix we prove the existence of the semi-invariants which are Plucker coordinates in the corresponding projective spaces. Then we use these semi-invariants to constr…

2013-12-16math-phmath.MPnlin.SI

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Generalized N=4 supersymmetric Toda lattice hierarchy and N=4 superintegrable mapping

It is shown that the one-dimensional generalized N=4 supersymmetric Toda lattice (TL) hierarchy (nlin.Si/0311030) contains the N=4 super-KdV hierarchy with the first flow time in the role of space coordinate. Two different N=2 superfield forms of the generalized N=4 supersymmetric TL equation, which are useful when solving the N=4 super-KdV and (1,1)-GNLS hierarchies, are discussed.

2007-01-16hep-thmath-phmath.MP

A note on the superintegrability of the Toda lattice

The superintegrability of the non-periodic Toda lattice is explained in the framework of systems written in action-angles coordinates. Moreover, a simpler form of the first integrals is given.

2006-06-30nlin.SI

Explicit Semi-invariants and Integrals of the Full Symmetric sl(n) Toda Lattice

We show how to construct semi-invariants and integrals of the full symmetric sl(n) Toda lattice for all n. Using the Toda equations for the Lax eigenvector matrix we prove the existence of semi-invariants which are homogeneous coordinates in the corresponding projective spaces. Then we use these semi-invariants to construct the integrals. The existence of additional integrals which constitute a full set of independent non-involutive integrals was known but the chopping and Kostant procedures have crucial computational complexities already for low-rank Lax matrices and are practically not appli…

2013-12-18hep-thmath-phmath.MP

Bihamiltonian geometry and separation of variables for Toda lattices

We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifolds, we show that these systems can be explicitly integrated via the classical Hamilton-Jacobi method in the so-called Darboux-Nijenhuis coordinates.

2000-02-08nlin.SI

Lusztig Factorization Dynamics of the Full Kostant-Toda Lattices

We study extensions of the classical Toda lattices at several different space-time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant-Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel Lie algebras. Our study brings into play factorizations of Loewner-Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different…

2023-03-10nlin.SI

Canonical transformations of the extended phase space, Toda lattices and Stackel family of integrable systems

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 Stackel systems.

1999-09-10nlin.SIsolv-int

A non-local Poisson bracket for Coxeter--Toda lattices

We present a non-local Poisson bracket defined on the phase space $G^{u,v}/H$ of a Coxeter--Toda lattice, where $G^{u,v}$ is a Coxeter double Bruhat cell of $GL_n$ and $H$ is the subgroup of diagonal matrices. This non-local Poisson bracket is given in an appropriate set of coordinates of $G^{u,v}/H$ derived from the so-called {\it factorization parameters}. We prove that the generalized Bäcklund--Darboux transformations $σ_{u,v}^{u',v'}: G^{u,v}/H\to G^{u',v'}/H$ are Poisson maps. We exploit that fact to show that the non-local Poisson bracket corresponds to the Atiyah--Hitchin bracket under …

2018-09-22nlin.SI

On the geometry of certain isospectral sets in the full Kostant-Toda lattice

We use momentum mappings on generalized flag manifolds and their momentum polytopes to study the geometry of the level sets of the 1-chop integrals of the full Kostant-Toda lattice in certain isospectral submanifolds of the phase space. We derive expressions for these integrals in terms of Plücker coordinates on the flag manifold in the case that all eigenvalues are zero and compare the geometry of the base locus of their level set varieties with the corresponding geometry for distinct eigenvalues. Finally, we illustrate and extend our results in the context of the full sl(3,C) and sl(4,C) Kos…

1996-06-19nlin.SIsolv-int