Discrete Toda Field Equations

We study an integrable system related to the relativistic Toda lattice. The bilinear representation of this lattice is given and the B\"ackulund transformation obtained. A fully discrete version is also introduced with its bilinear B\"acklund transformation and Lax pair. One-soliton solution of the discrete system is presented by use of B\"acklund transformation.

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.

New integrable lattice systems are introduced, their different integrable discretization are obtained. B\"acklund transformations between these new systems and the relativistic Toda lattice (in the both continuous and discrete time formulations) are established.

We propose a new integrable generalization of the Toda lattice wherein the original Flaschka-Manakov variables are coupled to newly introduced dependent variables; the general case wherein the additional dependent variables are vector-valued is considered. This generalization admits a Lax pair based on an extension of the Jacobi operator, an infinite number of conservation laws and, in a special case, a simple Hamiltonian structure. In fact, the second flow of this generalize...

The two-dimensional quantum lattice Toda model for the affine and simple Lie algebras of the type A is considered. For its known L-operator a correction of the second order in the lattice parameter is found. It is proved that the equation determining a correction of the third order has no solutions.

A new parameterisation of the solutions of Toda field theory is introduced. In this parameterisation, the solutions of the field equations are real, well-defined functions on space-time, which is taken to be two-dimensional Minkowski space or a cylinder. The global structure of the covariant phase space of Toda theory is examined and it is shown that it is isomorphic to the Hamiltonian phase space. The Poisson brackets of Toda theory are then calculated. Finally, using the me...

For any classical Lie algebra $g$, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers $(m,n)$. The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for $\mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,n\leq3$ are also given. For all $m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Toda chains are natural reductions of that of th...

We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and $E_6$ by Goodman and Wallach at the beginning of the 1980's. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types.

The paper deals with affine 2-dimensional Toda field theories related to simple Lie algebras of the classical series ${\bf D}_r$. We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of $\mathfrak{g}$. This is illustrated on the examples ${\bf D}_{r+1}^{(1)} \to {\bf B}_r^{(1)}$ and ${\bf D}_4^{(1)} \to {\bf G}_2^{(1)}$ using external automorphisms of the corresponding extend...

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several well-known (1+1) and one-dimensional lattices is discussed. In particular, we consider a non-commutative analog of the scheme: discrete Toda equations $\rightarrow$ Somos-$N$ sequences $\rightarrow$ discrete Painlev\'e equations.