On the full Kostant-Toda system and the discrete Korteweg-de Vries equations

The relation between the solutions of the full Kostant–Toda lattice and the discrete Korteweg–de Vries equation is analyzed. A method for constructing solutions of these systems is given. As a consequence of the matricial interpretation of this method, the transform of Darboux is extended for general Hessenberg banded matrices

Complex high order Toda and Volterra lattices

Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of B¨acklund transformations in such a way that each new generalized Toda solution is related to a generalized Volterra solution.Dirección General de Investigación, Ministerio de Educación y Ciencia, MTM2006-13000-C03-02; Universidad Politécnica de Madrid; Comunidad Autónoma de Madrid CCG06-UPM/MTM- 539; CMUC/FC

Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems

The Darboux transformation and the complex Toda lattice

Abstract It is well known that each solution of the Toda lattice can be represented by a tridiagonal matrix J(t). Under certain restrictions, it is possible to obtain some new solution by using the Darboux transformation of J(t) − CI. Our goal is the extension of this fact, which is known for the real lattice, to high order complex Toda lattices as well as to the bi-infinite Toda lattice. In this latter case, we use the factorization LU for block-tridiagonal matrices. The Toda lattice We study the construction of some solutions { from another given solution {α n (t) , λ n (t)} , n ∈ Z. We consider: 1. the semi-infinite problem: S = N, λ 1 = 0, 2. the infinite problem: In [6] the semi-infinite complex problem was analyzed. In the real, infinite case, sufficient conditions for the existence of a new solution were given in The problem: obtain a similar result to the complex infinite Toda lattice. The generalized Toda lattice In a more general way, when S = N we consider the generalized Toda where we denote by J i,j (t) (respectively J p i,j (t)) the entry in the (i+1)-row and (j + 1)-column of matrix J(t) (respectively (J(t)) The generalized Toda lattice admits a Lax pair representation, i.e. a formulation in terms of the commutator of two operators, J(t) = [J(t), K(t)] = J(t)K(t) − K(t)J(t) , where we prove the existence of is another solution of (2), and Γ(t) is a solution of the Volterra lattice: 3 Relation between the generalized Toda lattice and some polynomials The matrix J(t) t defines the sequence of polynomials given by The main tools in the proof of [2, Th. 1.3]: a. We have established the dynamic behavior of P n (t, z), where C ∈ C verifies (3). The sequence Q (C) n (t, C) satisfies a threeterm recurrence relation whose coefficients define the new generalized solution J(t) = J(t, C) The new solutions and the Darboux transformation If we define and C ∈ C verifies (3), then there exist The new solution is defined by the Darboux transformation of J (1) (t) − CI, this is, The infinite Toda lattice Let us consider (1) with S = Z and take the infinite matrix The infinite Toda lattice admits also a Lax pair representation. However, in this case it is not possible to use directly the sequences of polynomials associated to J. , n ∈ N , it is possible to change the infinite recurrence relation to a semi-infinite recurrence relation, where E m , V m , m ∈ N , are 2 × 2-finite matrices. In this way, we can study the infinite case as a semi-infinite vectorial case. The vectors R n are not polynomials, but we can prove where the sequence {C n } of 2 × 2 matrices verifies and for each i = 1, 2, 3, 4, c ni is a polynomial in z , deg c ni ≤ n − 1. Taking I −1 := 1 0 0 −1 , W n := I −1 V n , n ∈ N , we can shoẇ , n = 2, 3, . . . This is, {W n , E n } is a solution of a semi-infinite matricial Toda lattice, like (1). The infinite Toda lattice and the Darboux transformation We defin

Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non-symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems

Matrix interpretation of multiple orthogonality

In this work we give an interpretation of a (s(d + 1) + 1)-term recurrence relation in terms of type II multiple orthogonal polynomials.We rewrite this recurrence relation in matrix form and we obtain a three-term recurrence relation for vector polynomials with matrix coefficients. We present a matrix interpretation of the type II multi-orthogonality conditions.We state a Favard type theorem and the expression for the resolvent function associated to the vector of linear functionals. Finally a reinterpretation of the type II Hermite- Padé approximation in matrix form is given

Orthogonal polynomial interpretation of q-Toda and q-Volterra equations

The correspondences between dynamics of q-Toda and q-Volterra equations for the coefficients of the Jacobi operator and its resolvent function are established. The orthogonal polynomials associated with these Jacobi operators satisfy an Appell condition, with respect to the q-difference operator Dq . Lax type theorems for the point spectrum of the Jacobi operators associated with these equations are obtained. Examples related with the big q-Legendre, discrete q-Hermite I, and little q-Laguerre orthogonal polynomials and q-Toda and q-Volterra equations are given