Orlando Ragnisco

Integrable symplectic maps

Abstract In this paper, we first give a terse survey of symplectic maps, their canonical formulation and integrability. Then, we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices. A number of illustrative examples are provided.

R-Matrices and Higher Poisson Brackets for Integrable Systems

The tri-hamiltonian nature of Lax-equations is revealed: starting with an R-matrix on an associative algebra g equipped with a trace form there are g compatible Poisson brackets with linear, quadratic and cubic dependence on the coordinates. The invariant functions (Casimir functions) on g* are in involution relative to these brackets, they yield a hierarchy of integrable tri-hamiltonian Lax-equations. The results can be applied to solvable PDE’s such as the Korteweg-de Vries equation as well as to finite integrable systems such as the Toda lattice. In these cases the Poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well-known recursion operators.

Integrable time-discretisation of the Ruijsenaars-Schneider model

An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 XYZ Heisenberg magnet. We present a Lax pair, the sympletic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.

Reduction techniques for infinite dimensional Hamiltonian systems: some ideas and applications

In the language of tensor analysis on differentiable manifolds, we present a reduction method of integrability structures, and apply it to recover some well-known hierarchies of integrable nonlinear evolution equations.

A SYSTEMATIC CONSTRUCTION OF COMPLETELY INTEGRABLE HAMILTONIANS FROM COALGEBRAS

A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir elements is presented. In particular, this construction shows that quantum deformations can be interpreted as generating structures for integrable deformations of Hamiltonian systems with coalgebra symmetry. In order to illustrate this general method, the algebra and the oscillator algebra are used to derive new classical integrable systems including a generalization of Gaudin-Calogero systems and oscillator chains. Quantum deformations are then used to obtain some explicit integrable deformations of the previous long-range interacting systems and a (non-coboundary) deformation of the (1 + 1) Poincare algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.

MASTERSYMMETRIES, ANGLE VARIABLES, AND RECURSION OPERATOR OF THE RELATIVISTIC TODA LATTICE

Conserved quantities, bi‐Hamiltonian formulation, and recursive structure of the relativistic Toda lattice (RT) are obtained in an algorithmic way without making use of the Lax representation. Furthermore, for the multisoliton solutions the gradients of the angle variables are described in terms of mastersymmetries. A new hierarchy of completely integrable systems is discovered, which turns out to correspond to the ‘‘negative’’ of the hierarchy of RT. Thus it is shown that the full algebra of time‐dependent symmetry group generators for each member of the RT hierarchy is isomorphic to the algebra of first order differential operators with Laurent polynomials as coefficients. The surprising phenomenon is revealed that the members of the RT hierarchy are connected to their negative counterparts by explicit Backlund transformation.

Recursion operator and bäcklund transformations for the Ruijsenaars-Toda lattice

Abstract We exhibit the recursion operator and the whole class of Backlund transformations for a relativistic version of the Toda lattice recently introduced by Ruijsenaars. These results allow us to prove the complete integrability of the system.

LAX REPRESENTATION AND COMPLETE-INTEGRABILITY FOR THE PERIODIC RELATIVISTIC TODA LATTICE

Abstract Three different Lax representations for the periodic relativistic Toda lattice are exhibited. The complete integrability of the system is also proven.

Continuous and discrete matrix Burgers’ hierarchies

SummaryWe derive two hierarchies of matrix nonlinear evolution equations which reduce to the Burgers’ hierarchy in the scalar case and can be linearized by a matrix analogue of the Hopf-Cole transformation: for these hierarchies we display the associated class of Bäcklund transformations and show some special kinds of explicit solutions. More-over, by exploiting a discrete version of the Hopf-Cole transformation, we are also able to construct two hierarchies of linearizable nonlinear difference evolution equations and to derive for them Bäcklund trans-formations and explicit solutions.RiassuntoIn questo lavoro si derivano due gerarchie di equazioni di evoluzione nonlineari matriciali che possono essere linearizzate mediante un analogo matriciale della trasformazione di Hopf-Cole e si riducono nel caso scalare alla già nota gerarchia di Burgers. Per queste due gerarchie, come pure per le loro versioni discrete (anch’esse linearizzabili) si ottengono le trasformazioni di Bäcklund e si mostrano alcuni tipi significativi di soluzioni esplicite.

A novel hierarchy of integrable lattices

In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattices, whose continuum limit is the AKNS hierarchy. In contrast to other differential-difference versions of the AKNS system, our hierarchy is endowed with a canonical Poisson structure and, moreover, it admits a vector generalization. We also solve the associated spectral problem and explicitly construct action-angle variables through the r-matrix approach.In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattice, whose continuum limit is the AKNS hierarchy. In contrast with other differential-difference versions of the AKNS system, our hierarchy is endowed with a canonical Poisson structure and, moreover, it admits a vector generalisation. We also solve the associated spectral problem and explicity contruct action-angle variables through the r-matrix approach.