On perturbations of the periodic Toda lattice

A class of Hamiltonian systems including perturbations of the periodic Toda lattice and homogeneous cosmological models is studied. Separatrix approximation of oscillation regimes in these systems connected with Coxeter groups is obtained. Hamiltonian systems connected with simple Lie algebras are pointed out, which generalize the system describing periodic Toda lattice and allow theL -A pair representation.     

Poisson geometry of the analog of the Miura maps and Bäcklund-Darboux transformations for equations of Toda type and periodic Toda flows

We show that the analog of the Miura maps and Bäcklund-Darboux transformations for a general class of equations of Toda type and for a generalized class of periodic Toda flows are isomorphisms of Poisson Lie groups.     

On the bi-Hamiltonian structure of Toda and relativistic Toda lattices

We introduce a general quadratic Poisson bracket on the associative algebra equipped with non-degenerate scalar product. With the help of this bracket we obtain the interpretation of the Toda and relativistic Toda lattices as the restrictions of one and the same bi-Hamiltonian system to two different low-dimensional manifolds, which are Poisson submanifolds with respect to two brackets simultaneously.     

Two-dimensional generalized Toda lattice

The zero curvature representation is obtained for the two-dimensional generalized Toda lattices connected with semisimple Lie algebras. The reduction group and conservation laws are found and the mass spectrum is calculated.     

Compatible Lie algebroids and the periodic Toda lattice

We give the set of maps from to the structure of a Poisson manifold endowed with a pair of compatible Lie algebroids. A suitable reduction process, of the Marsden–Ratiu type, yields a smaller manifold with the same geometrical properties as the original manifold. Moreover, is a bi-Hamiltonian manifold and the flows naturally defined on it are the periodic Toda flows.     

Supersymmetric two-dimensional Toda lattice

The two-dimensional Toda lattice connected with contragradient Lie superalgebras is studied. The systems of linear equations associated with the models for which the inverse scattering method is applicable are written down. The reduction group is calculated.     

Toda lattice hierarchy and conservation laws

Time evolutions of the Toda lattice hierarchies of Ueno and Takasaki are induced by Hamiltonians which are conservation laws for the original (one and two dimensional) Toda lattice obtained by Olive and Turok. Moreover these Hamiltonians for two dimensional Toda lattice hierarchy are also conserved quantities of the two component KP hierarchy in which that system is embedded. The one dimensional Toda lattice hierarchy is characterized by the bilinear relations, and a new version of the one dimensional Toda lattice hierarchy is constructed. Generalized Toda lattice hierarchies associated to all affine Lie algebras are presented.     

Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation

In this paper, we construct Hamiltonian systems for 2 N particles whose force depends on the distances between the particles. We obtain the generalized finite nonperiodic Toda equations via a symmetric group transformation. The solutions of the generalized Toda equations are derived using the tau functions. The relationship between the generalized nonperiodic Toda lattices and Lie algebras is then be discussed and the generalized Kac-van Moerbeke hierarchy is split into generalized Toda lattices, whose integrability and Darboux transformation are studied.     

Higher Toda Mechanics and Spectral Curves

For each of the Lie algebras gln and gl～n., we construct a family of integrable generalizations of the Toda chains characterized by two integers m and m-. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m = m-,we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic (m ,m-)-Toda chains, which turns out to be very different for different pairs of m and m-. Finally we also obtain thenonabelian generalizations of the (m , m-)-Toda chains in an explicit form.     

Algebraic structure of Toda systems

Following the Leningrad school an operator $\text{P}$ is constructed which guarantees the classical complete integrability of the Toda molecule and Toda lattice equations. This quantity depends in a uniform way upon the root system of the underlying algebra, respectively the simple Lie algebras and the affine euclidean Kac-Moody algebras.     

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

The noncommutative Toda hierarchy is studied with the help of Moyal deformation by a reduction on the non-commutative two dimensional Toda hierarchy. Further we generalize the noncommutative Toda hierarchy to the extended noncommutative Toda hierarchy. To survey on its integrability, we construct the bi-Hamiltonian structure and noncommutative conserved densities of the extended noncommutative Toda hierarchy by means of the R-matrix formalism. This extended noncommutative Toda hierarchy can be reduced to the extended multicomponent Toda hierarchy, extended Z_N?-Toda hierarchy, extended Toda hierarchy respectively by reductions on Lie algebras.     

Higher Toda Mechanics and Spectral Curves

For each of the Lie algebras gln and g~ln we construct a family of integrable generalizations of the Toda chains characterized by two integers m and m_. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m =m_,we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic (m ,m_)-Toda chains, which turns out to be very different for different pairs of m and m_. Finally we also obtain the nonabelian generalizations of the (m ,m_)-Toda chains in an explicit form.     

Riccati-Type Equations, Generalised WZNW Equations, and Multidimensional Toda Systems

We associate to an arbitrary ℤ-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional extension of these equations is given. The generalisation of the associated Redheffer–Reid differential systems appears in a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established. Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given. Received: 3 August 1998 / Accepted: 21 December 1998     

Drinfeld-Sokolov Construction of Heterotic Toda Model

A free field representation for the left-right asymmetric conformal Toda theory based on simplf-laced even-rank Lie algebras is given. It is shown that the classical chiral exchange algebra for such theories can be reconstructed from free chiral bosons via Drinfeld-Sokolov linear systems, and is a bit more complicated than that of the standard Toda due to some additional δ-function terms and extra degrees of freedom.     

The quantum mechanical Toda lattice,II

The periodic Toda lattice consists of N particles which move along a closed line and are coupled with an exponential spring to their immediate neighbors. This system, in contrast to the open Toda lattice, has only bound states. In the method of Kac and Van Moerbeke, the classical periodic Toda chain is transformed to a new of set of canonically conjugate variables, μ and ν, which are closely related to the natural coordinates of an open Toda chain with one particle less. The quantum mechanical eigenfunctions for this reduced system are constructed explicitly, and this allows the quantum mechanical analogs of μ and ν to be defined. The bounds states for the periodic Toda chain are then written as linear combinations of functions resembling the wave functions of the reduced open chain. These functions satisfy a set of remarkably simple recursion formulas, and the coefficients in the expansion can be written essentially as a product of as many factors as pairs of conjugate variables μ and ν. Each factor is given as a solution of a second order difference equation which can be recognized as a quantum analog for the equations of motion of one pair μ and ν. The quantization conditions result from cancelling out the exponential growth in the overall wave function, and are phrased in terms of certain phase angles being submultiples of π according as the representation of the group of cyclic permutations. The calculations are simple for N = 3, and moderately tricky for N = 4 although the results are always fairly obvious.     

The finite Toda lattices

Connection is established between one-dimensional Toda lattices, constructed on the basis of the systems of simple roots of classical and affine Lie algebras, and other integrable systems of interacting particles. That connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends. Some of the new lattices admit non-Abelian generalizations.     

Aspects of affine Toda field theory

Several of the recently discovered classical and quantum features of affine Toda field theory are briefly reviewed, with particular emphasis on the Lie algebraic structure of masses, conserved quantities and S-matrices.     

Non-compact symmetric spaces and the Toda molecule equations

It has been shown by Olshanetsky and Perelomov that the Toda molecule equations associated with any Lie groupG describe special geodesic motions on the Riemannian non-compact symmetric space which is the quotient of the normal real form ofG, G ^N, by its maximal compact subgroup. This is explained in more detail and it is shown that the fundamental Poisson bracket relation involving the Lax operatorA and leading to the Yang-Baxter equation and integrability properties is a direct consequence of the fact that the Iwasawa decomposition forG ^N endows the symmetric space with a hidden group theoretic structure.Supported by CNP _q (Brasil)     

T-duality in Affine NA Toda Models

The construction of non-Abelian affine Toda models is discussed in terms of its underlying Lie algebraic structure. It is shown that a subclass of such non-conformal two-dimensional integrable models naturally leads to the construction of a pair of actions, which share the same spectra and are related by canonical transformations.     

A trihamiltonian extension of the Toda lattice

A new Poisson structure is defined on a subspace of the Kupershmidt algebra, isomorphic to the space H$\mathfrak{H}$ of n×n$n\times n$ Hermitian matrices. The new Poisson structure is of Lie–Poisson type with respect to the standard Lie bracket of H$\mathfrak{H}$. This Poisson structure (together with two already known ones, obtained through a r$r$-matrix technique) allows to construct an extension of the periodic Toda lattice with n$n$ particles that fits in a trihamiltonian recurrence scheme. Some explicit examples of the construction and of the first integrals found in this way are given.