Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice. Résumé Deux formalismes différents pour étudier les sym...

We give a complete proof of the result stated in [1], that the general Einstein metric with a symmetry, an anti-self-dual Weyl tensor and nonzero scalar curvature is determined by a solution of the SU(∞)Toda field equation. We consider the two canonical forms found for solutions to the same problem by Przanowski [2] and show that his Class A will reduce to the Toda equation with respect to a se...

We present results on the relationships between persistent currents and the known conservation laws in the classical Toda ring. We also show that perturbing the integrability leads to a decay of the currents at long times with a time scale that is determined by the perturbing parameter. We summarize several known results concerning the Toda ring in one dimension, and present new results relatin...

We consider the ǫ-regime of QCD in 3 dimensions. It is shown that the leading term of the effective partition function satisfies a set of Toda lattice equations, recursive in the number of flavors. Taking the replica limit of these Toda equations allows us to derive the microscopic spectral correlation functions for the QCD Dirac operator in 3 dimensions. For an even number of flavors we reprod...

The Toda lattice hierarchy is discussed in connection with the topological description of the c = 1 string theory compactified at the selfdual radius. It is shown that when special constraints are imposed on the Toda hierarchy, it reproduces known results of the c = 1 string theory, in particular the W1+∞ relations among tachyon correlation functions. These constraints are the analogues of stri...

The bootstrap equations for the ADE series of purely elastic scattering theories have turned out to be intimately connected with the geometry of root systems and the Coxeter element. An informal review of some of this material is given, mentioning also a couple of other contexts – the Pasquier models, and the simply-laced affine Toda field theories – where similar structures are encountered. Th...

We consider tt∗ equations appearing in the study of localized tachyon condensations. They are described by various Toda system when we consider the condensation by the lowest tachyon corresponding to the monomial xy. The tachyon potential is calculated as a solution to these equations. The Toda system appearing in the deformation of C/Zn by xy is identical to that of Dn singularity deformed by ...

The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov–Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and Toda pictures of the dispersionless Wh...

In previous work the author found solutions to the Toda equations that were expressed in terms of determinants of integral operators. Here it is observed that a simple variant yields solutions to the matrix Toda equations. As an application another derivation is given of a differential equation of Sato, Miwa and Jimbo for a particular Fredholm determinant. During the last twenty years, beginnin...

The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integrals in involution is studied. It is shown that these systems can be considered as generalizing the semi-infinite Toda lattice which is an example of a bidiagonal system, a case to which special attention is given. The generic coadjoint orbits of the Banach Lie group of bidiagonal bounded operators are...