The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras (original) (raw)

Complexiton solutions of the Toda lattice equation

Physica A: Statistical Mechanics and its Applications, 2004

A set of coupled conditions consisting of differential-difference equations is presented for Casorati determinants to solve the Toda lattice equation. One class of the resulting conditions leads to an approach for constructing complexiton solutions to the Toda lattice equation through the Casoratian formulation. An analysis is made for solving the resulting system of differential-difference equations, thereby providing the general solution yielding eigenfunctions required for forming complexitons. Moreover, a feasible way is presented to compute the required eigenfunctions, along with examples of real complexitons of lower order.

W-geometry of the Toda systems associated with non-exceptional simple Lie algebras

Communications in Mathematical Physics, 1996

The present paper describes the W -geometry of the Abelian finite non-periodic (conformal) Toda systems associated with the B, C and D series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Plücker embedding of the Acase to the flag manifolds associated with the fundamental representations of B n , C n and D n , and a direct proof that the corresponding Kähler potentials satisfy the system of two-dimensional finite non-periodic (conformal) Toda equations. It is shown that the W -geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) of CP N with appropriate choices of N . In addition, these W-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Plücker embedding. These conditions are automatically fulfiled when Toda equations hold.

Point symmetries of generalized Toda field theories: II. Symmetry reduction

Journal of Physics A: Mathematical and General, 2000

A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two "coupling" matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point symmetries of these theories are found for an infinite, semi-infinite and finite number of fields. Special attention is accorded to conformal invariance and its breaking.

Explicit solutions of classical generalized toda models

Inventiones Mathematicae, 1979

The explicit solution of the generalized Toda models connected with the root systems of simple Lie algebras is constructed. This solution is obtained by a special projection of geodesic flows on some symmetric spaces on the phase space of the Toda models. In particular, for the usual N-body Toda model the solutions are expressed by minors of the exponent of the Jacobi matrix depending on initial data. * This paper has been published early as a preprint ITEP-157 (1978)

The complex Toda chains and the simple Lie algebras - solutions and large time asymptotics

Journal of Physics A: Mathematical and General, 1998

The asymptotic regimes of the N -site complex Toda chain (CTC) with fixed ends related to the classical series of simple Lie algebras are classified. It is shown that the CTC models have much richer variety of asymptotic regimes than the real Toda chain (RTC). Besides asymptotically free propagation (the only possible regime for the RTC), CTC allow bound state regimes, various intermediate regimes when one (or several) group(s) of particles form bound state(s), singular and degenerate solutions. These results can be used e.g., in describing the soliton interactions of the nonlinear Schrödinger equation. Explicit expressions for the solutions in terms of minimal sets of scattering data are proposed for all classical series B r -D r .

Reduction of Toda Lattice Hierarchy to Generalized KdV Hierarchies and the Two-Matrix Model

1995

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 an 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 W M +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.

Generalized Casorati Determinant and Positon–Negaton-Type Solutions of the Toda Lattice Equation

Journal of the Physical Society of Japan, 2004

A set of conditions is presented for Casorati determinants to give solutions to the Toda lattice equation. It is used to establish a relation between the Casorati determinant solutions and the generalized Casorati determinant solutions. Positons, negatons and their interaction solutions of the Toda lattice equation are constructed through the generalized Casorati determinant technique. A careful analysis is also made for general positons and negatons, the resulting positons and negatons of order one being explicitly computed. The generalized Casorati determinant formulation for the two dimensional Toda lattice (2dTL) equation is presented. It is shown that positon, negaton and complexiton type solutions in the 2dTL equation exist and these solutions reduce to positon, negaton and complexiton type solutions in the Toda lattice equation by the standard reduction procedure.