Frank Nijhoff

gt; Fermi National Accelerator Laboratory

Hierarchies of integrable nonlinear evolution equations and Wess-Zumino Lagrangians

Algebraic Structure of Integrable Systems in D= 2+ 1 and Routes Towards Multidimensional Integrabilitv

D-Algebras, the D-Simplex Equations, and Multidimensional Integrability

Series on Knots and Everything, 1993

Recently R . Lawrence has introduced axioms for a category of d-algebras , i.e. algebras with d-a... more Recently R . Lawrence has introduced axioms for a category of d-algebras , i.e. algebras with d-ary operations . The natural associativities are defined from geometrical considerations of moves in simplex partitions . We study these algebras in connection with the tetrahedron and higher simplex equations that were proposed by Zamolodchikov and descendants a decade ago. These are the natural generalizations of the quantum Yang-Baxter equations . We discuss the connection of these algebras with the issue of integrability in multi-dimensions.

Integrabile Quantum Mappings and Quantization Aspects of Integrable Discrete-Time Systems

arXiv: High Energy Physics - Theory, 1993

We study a quantum Yang-Baxter structure associated with non-ultralocal lattice models, and discu... more We study a quantum Yang-Baxter structure associated with non-ultralocal lattice models, and discuss the canonical structure of a class of integrable quantum mappings, i.e. canonical transformations preserving the basic commutation relations. As a particular class of solutions we present two examples of quantum mappings associated with the lattice analogues of the KdV and MKdV equations, together with their exact quantum invariants.

Integrable Lattice Equations

Integrable Lattice systems are space-and time-discretizations of integrable partial differential ... more

Integrable lattice version of the massive thirring model and its linearization

Physics Letters A, 1983

ABSTRACT

Linearizing integral transform and partial difference equations

Physics Letters A, 1984

A linearizing integral transform is proposed which relates solutions of a spectral problem associ... more A linearizing integral transform is proposed which relates solutions of a spectral problem associated with a class of integrable partial difference equations to any given solution of the spectral problem. Examples of this class are lattice versions of the isotropic Heisenberg spin chain, the nonlinear Schrödinger equation and the (complex) sine-Gordon equation.

A TWO-PARAMETER ELLIPTIC EXTENSION OF THE LATTICE KdV SYSTEM

NATO Science Series

ABSTRACT In [1] we presented a novel integrable lattice system given by the following coupled sys... more

Integrable Mappings and Soliton Lattices

Solitons are found as solutions of integrable partial differential equations (PDE’s) which have b... more Solitons are found as solutions of integrable partial differential equations (PDE’s) which have been extensively studied during the last three decades. More recently there has been increasing interest in partial difference equations with fields defined at the sites of 2- and 3-dimensional (2D and 3D) lattices. They are the discrete-time analogues of PDE’s as well as of partial difference equations depending on time as well as on 1 or 2 discrete variables. During the last years a variety of integrable 2D and 3D lattices has been found, by the direct linearization method (DLM) which is based on a linear integral equation with arbitrary measure and contour [1]–[4], as well as by the bilinearization method [5] and the τ-function approach [6]. By taking continuum limits the lattice equations yield hierarchies of integrable partial difference equations (or PDE’s) with one or more continuous variables together with a Poisson structure and an infinite number of conserved quantities in involution [7,8], see also ref. [9].

Statistical mechanics, soliton theory, and nonlinear dynamics-In honour of HW Capel on the occasion of his 60th birthday-Preface

On a (2+1)-dimensional version of the Krichever-Novikov equation

Physics Letters A, 1991

Abstract A (2+1)-dimensional version of the Krichever-Novikov equation is studied. This equation ... more

Integrable mappings of KdV type and hyperelliptic addition formulae

Symmetries and Integrability of Difference Equations

Discrete Painlevé Equations

The Painlevé Property, 1999

The direct linearizing transform for the τ function in three-dimensional lattice equations

Physics Letters A, 1985

Abstract A connection between the direct linearization and the τ function for solutions of the th... more Abstract A connection between the direct linearization and the τ function for solutions of the three-dimensional lattice Toda equation is given. The connection is via a fermionic path integral containing an action defined in terms of the kernel of the linearizing integral transformation for the Toda equation. General properties of the τ function are derived, and some relations with the representation theory of infinite-dimensional Lie algebras are mentioned.

Lattice equations associated with the Landau-Lifschitz equations

Physics Letters A, 1989

ABSTRACT Equations with an elliptic pencil are studied which can be considered to constitute the ... more ABSTRACT Equations with an elliptic pencil are studied which can be considered to constitute the (Darboux) Bäcklund transformations for the Landau-Lifschitz (LL) equation. The compatibility conditions yield a system that gives rise to a discrete version of the LL equations on a two-dimensional lattice. A continuum limit allows one to recover the hierarchy of LL equations from this system.

Bäcklund transformations and linearization of the anisotropic Heisenberg spin chain

Physics Letters A, 1982

Bäcklund transformations for a class of integrable partial differential equations are shown to ar... more Bäcklund transformations for a class of integrable partial differential equations are shown to arise from singular transformations of measures in the associated linear integral equation. As a corollary a new linear integral equation is obtained for the solutions of the classical Heisenberg spin chain with uniaxial anisotropy.

Direct linearization of nonlinear difference-difference equations

Physics Letters A, 1983

ABSTRACT

Linearization of nonlinear differential-difference equations

Physica D: Nonlinear Phenomena, 1984

ABSTRACT

The direct linearizing transform for three-dimensional lattice equations

Physica D: Nonlinear Phenomena, 1986

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