Rational recursion operators for integrable differential-difference equations

In this paper we introduce the concept of pre-Hamiltonian pairs of difference operators, demonstrate their connections with Nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential {diference equations. We begin with a rigorous setup of the problem in terms of the skew field of rational (pseudo{diference) operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational (pseudo{diference) operators. In particular, we give a criteria for a rational operator to be weakly nonlocal. A difference operator is called pre-Hamiltonian, if its image is a Lie subalgebra with respect to the Lie bracket on the difference field. Two pre-Hamiltonian operators form a pre-Hamiltonian pair if any linear combination of them is preHamiltonian. Then we show that a pre-Hamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a pre-Hamiltonian pair. This provides a systematic method to check whether a rational operator is Nijenhuis. As an application, we construct a hamiltonian pair and thus a Nijenhuis recursion operator for the diferential{difference equation recently discovered by Adler & Postnikov. The Nijenhuis operator obtained is not weakly nonlocal. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well-known examples including the Toda, the Ablowitz{Ladik and the Kaup{Newell differential difference equations

Integrable systems, random matrices and applications

Nonlinear integrable systems emerge in a broad class of different problems in Mathematics and Physics. One of the most relevant characterisation of integrable systems is the existence of an infinite number of conservation laws, associated to integrable hierarchies of equations. When nonlinearity is involved, critical phenomena may occur. A solution to a nonlinear partial differential equation may develop a gradient catastrophe and the consequent formation of a shock at the critical point. The approach of differential identities provides a convenient description of systems affected by phase transitions, identifying a suitable nonlinear equation for the order parameter of the system. This thesis is aimed to give a contribution to the perspective offered by the approach of differential identities. We discuss how this method is particularly useful in treating mean-field theories, with some explicit application. The core of the work concerns the Hermitian matrix ensemble and the symmetric matrix ensemble, analysed in the context of integrable systems. They both underlie a discrete integrable structure in form of a lattice, satisfying a discrete integrable hierarchy. We have studied a particular reduction of both system and determined the continuum limit of the dynamics of the field variables at the leading order. Particular emphasis has been given to the study of the symmetric matrix ensemble. We have unveiled an unobserved double-chain structure shared by the field variables populating the lattice structure associated to the ensemble. In the continuum limit of a particular reduction of the lattice, we have found a new hydrodynamic chain, a hydrodynamic system with infinitely many components. We have shown that the hydrodynamic chain is integrable and we have conjectured the form of the associated hierarchy. The new integrable hydrodynamic chain constitutes per se an interesting object of study. Indeed, it presents some properties that are different from those shared by the standard integrable hydrodynamic chains studied in literature

Integrable many-body systems of Calogero-Ruijsenaars type

A dolgozat a Calogero-Ruijsenaars típusú Liouville integrálható rendszerekkel kapcsolatos alábbi eredményeinket mutatja be: 1. A racionális Calogero-Moser rendszer spektrális koordinátáit szolgáltató explicit formula bizonyítása. 2. A trigonometrikus BC(n) Sutherland rendszer hatás-szög duálisának kidolgozása hamiltoni redukció alkalmazásával. 3. A trigonometrikus BC(n) Sutherland rendszer egy Poisson-Lie deformációjának levezetése hamiltoni redukció alkalmazásával. 4. A hiperbolikus BC(n) Ruijsenaars-Schneider-van Diejen rendszer Lax reprezentációjának kidolgozása. 5. Trigonometrikus és elliptikus Ruijsenaars-Schneider modellek konstrukciója a komplex projektív téren. Abstract: This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle duality for the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 3. We derive a Poisson-Lie deformation of the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 4. We construct a Lax pair for the hyperbolic BC(n) Ruijsenaars-Schneider-van Diejen system. 5. We present an explicit construction of compactified trigonometric and elliptic Ruijsenaars-Schneider systems

Polinomios biortogonales y sus generalizaciones: una perspectiva desde los sistemas integrables

La conexión existente entre los polinomios ortogonales y otras ramas de la matemática, la física o la ingeniería es verdaderamente asombrosa. Además, no hay mejor prueba de la utilidad de estos que el propio crecimiento, avance perpetuo y generalización en diversas direcciones de lo que se entendía por polinomio ortogonal en los albores de la teoría. Conforme el concepto se fue generalizando, también fueron evolucionando las técnicas para su estudio, algunas de estas claramente influenciadas por aquellas disciplinas matemáticas con las que iban surgiendo conexiones. La perspectiva que esta tesis adopta frente a los polinomios ortogonales es un ejemplo de este tipo de influencias, compartiendo herramientas y entrelazandose con la teoría de los sistemas integrables. Una posición privilegiada en esta tesis la ocuparían las matrices de Gram semi in nitas; cada cual asociada a una forma sesquilineal adaptada al tipo de biortogonalidad en cuestión. A estas matrices se les impondrán una serie de condiciones cuyo objeto sería el de garantizar la existencia y unicidad de las secuencias biortogonales asociadas a las mismas. El siguiente paso consistiría en buscar simetrías de estas matrices de Gram. Existen dos razones por las que este esfuerzo resulta ventajoso. En primer lugar, cada simetría encontrada podría traducirse en propiedades de las secuencias biortogonales, por ejemplo: una estructura Hankel de la matriz es equivalente a gozar de la recurrencia a tres términos de los polinomios ortogonales; la simetría propia de las matrices asociadas a pesos clásicos (Hermite, Laguerre, Jacobi) implica la existencia del operador diferencial lineal de segundo orden de que los polinomios clásicos son solución; etc..