V. Papageorgiou

Integrable mappings and nonlinear integrable lattice equations

Abstract Periodic initial value problems of time and space discretizations of integrable partial differential equations give rise to multi-dimensional integrable mappings. Using the associated linear spectral problems (Lax pairs), a systematic derivation is given of the corresponding sets of polynomial invariants. The level sets are algebraic varieties on which the trajectories of the corresponding dynamical systems lie.

Similarity reductions of integrable lattices and discrete analogues of the Painlevé II equation

Abstract Using the direct linearization method similarity reductions of integrable lattices are constructed providing discrete analogues of the Painleve II equation. The reduction is obtained by imposing an non-autonomous integrable constraint on the corresponding lattice equations, which are differential-difference analogue of the KdV and modified KdV equation. An isomonodromic deformation problem for these system is derived. It is shown how this system in an appropriate continuum limit reduces to the PII equation.

COMPLETE INTEGRABILITY OF LAGRANGIAN MAPPINGS AND LATTICES OF KDV TYPE

Abstract We present the Lagrangian and (time-discrete) Hamiltonian structures of lattice discretizations of the KdV equation, as well as of the associated finite-dimensional mappings that we derived earlier. Complete integrability in the sense of Liouville of these mappings is established by showing involutivity of a complete set of integrals of the discrete-time dynamics. Similar results hold for lattices and mappings related to the MKdV and Toda equations.

Integrable mappings derived from soliton equations

We derive a hierarchy of ibtegrable mappings (integrable ordinary difference equations) corresponding to solutions of the initial-value problem of an integrable partial difference equation with periodic initial data. For each n ϵ N this hierarchy contains at least one integrable mapping Rn→Rn. The integrals of these mappings are constructed using the Lax pair of the underlying partial difference equation. Our approach is illustrated for the integrable partial difference analogues of the sine-Gordon and the (modified) Korteweg-de Vries equations.

Isomonodromic deformation problems for discrete analogues of Painlevé equations

Abstract Starting from isospectral problems of two-dimensional integrable mappings, isomonodromic deformation problems for the new descrete versions of the Painleve I–III equations are constructed. It is argued that they lead to differential and q-difference deformations (“quantizations”) of the corresponding spectral curves.

Integrable lattices and convergence acceleration algorithms

Abstract We show that a well-known convergence acceleration scheme, the ϵ-algorithm, when viewed as a two-variable difference equation, is nothing but the discrete Korteweg-de Vries lattice equation. The complete integrability of the latter confers to the ϵ-algorithm interesting properties among which the singularity confinement is outstanding. In fact, this property is used in order to derive the generalizations of the ϵ-accelerator leading to the most general form of the ϱ-algorithm. A new acceleration algorithm based on the modified Korteweg-de Vries lattice equation is also derived.

On the integrability of systems of nonlinear ordinary differential equations with superposition principles

A new class of ‘‘solvable’’ nonlinear dynamical systems has been recently identified by the requirement that the ordinary differential equations (ODE’s) describing each member of this class possess nonlinear superposition principles. These systems of ODE’s are generally not derived from a Hamiltonian and are classified by associated pairs of Lie algebras of vector fields. In this paper, all such systems of n≤3 ODE’s are integrated in a unified way by finding explicit integrals for them and relating them all to a ‘‘pivotal’’ member of their class: the projective Riccati equations. Moreover, by perturbing two parametrically driven projective Riccati equations (thus making them nonsolvable in the above sense) evidence is discovered of chaotic behavior on the Poincare surface of section—in the form of sensitive dependence on initial conditions—near a boundary separating bounded from unbounded motion.

Yang-Baxter maps and symmetries of integrable equations on quad-graphs

A connection between the Yang-Baxter relation for maps and the multidimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multiparameter symmetry groups of the equations. We use the classification results by Adler, Bobenko, and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multifield integrable equations.

Linearization and solutions of the discrete Painlevé III equation

Abstract We present particular solutions of the discrete Painleve III (d-P III ) equation of rational and special unction (Bessel) type. These solutions allow us to establish a close parallel between this discrete equation and its continuous counterpart. Moreover, we propose an alternate form for d-P III and confirm its integrability by explicitly deriving its Lax pair.

Orthogonal Polynomial Approach to Discrete Lax Pairs for Initial Boundary-Value Problems of the QD Algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.