We prove separation of variables for the most general (Dn type) periodic Toda lattice with 2 × 2 Lax matrix. It is achieved by finding proper normalisation for the corresponding Baker-Akhiezer function. Separation of variables for all other periodic Toda lattices associated with infinite series of root systems follows by taking appropriate limits.

A classification of Hankel determinant solutions of the restricted Toda chain equations is presented through polynomial Ansatz for moments. Each solution corresponds to the Sheffer class orthogonal polynomials. In turn, these solutions are equivalent to solutions with separated variables in Toda chain. These solutions lead naturally to explicit Hankel determinants of some combinatorial numbers.

We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Toda hierarchy, which is of independent interest as it so...

We provide a proof of a formula conjectured in [1] for some coefficients relevant in the principal vertex operator construction of a simply-laced affine algebra ĝ. These coefficients are important for the study of the topological charges of the solitons of affine Toda theories, and the construction of representations of non-simply-laced ĝ and their associated Toda solitons.

The Bäcklund transformations for the relativistic lattices of the Toda type and their discrete analogues can be obtained as the composition of two duality transformations. The condition of invariance under this composition allows to distinguish effectively the integrable cases. Iterations of the Bäcklund transformations can be described in the terms of nonrelativistic lattices of the Toda type....

The present article reviews methods from spectral theory and algebraic geometry for finding explicit solutions of the Toda equation, namely for the N -soliton solution and quasi-periodic solutions. Along they way basic concepts like Lax pairs, associated hierarchies, and Bäcklund transformations for the Toda equation are introduced.

Contents 1. Introduction 1 2. Geometrical setting near a dilated catenoid 6 3. Jacobi-Toda system on the Catenoid 9 4. Jacobi operator and the linear Jacobi-Toda operator on the catenoid. 16 5. Approximation of the solution of the theorem 1 23 6. Proof of theorem 1. 30 7. gluing reduction and solution to the projected problem.

We establish locality estimates, known as Lieb–Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb–Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a finite velocity for certain perturbations of these systems.

In this paper, the relations between three special forms of matrix integrals and their associated integrable differential-difference systems are considered. It turns out that these matrix integrals with special β = 2 and 1, 4 satisfy the differential-difference KP equation, the two-dimensional Toda lattice, the semi-discrete Toda equation and their Pfaffianized systems, respectively.

The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand, to periodic systems on the other. Boundary conditions are introduced to reduce theories on an infinite lattice to those on semi-infinite, or finite ones.