Frank W. Nijhoff

The Discrete Korteweg—de Vries Equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.

Lax pair for the Adler (lattice Krichever-Novikov) System

Abstract In the paper Int. Math. Res. Notices 1 (1998) 1 a lattice version of the Krichever–Novikov equation was constructed. We present in this Letter its Lax pair and discuss its elliptic form.

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.

Direct linearization of nonlinear difference-difference equations

Abstract Starting from the linear integral equation for the solutions of the Korteweg-de Vries (KdV) equation, we obtain the direct linearization of a general nonlinear difference-difference equation. In a continuum limit this equation reduces to a general integrable differential-difference equation which contains e.g. the Toda equation and the discrete KdV and MKdV as special cases.

The discrete and continuous Painlevé VI hierarchy and the Garnier systems

We present a general scheme to derive higher-order members of the Painleve VI (PVI) hierarchy of ODEs as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.

Linear integral equations and nonlinear difference-difference equations

In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differential-difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference versions and the differential-difference versions of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrodinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation.

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.