# School of Mathematics

## Prof Frank W Nijhoff

Professor of Mathematical Physics

Applied Mathematics

### Contact details

Room: 9.08

Tel: +44 (0)113 3435120

Email: F.W.Nijhoff @ leeds.ac.uk

### Keywords

Mathematical Physics

Discrete and Continuous Integrable Systems

Quantum Integrability

Painleve Equations

Lagrangian Multiform Theory

Integrable Lattice Equations

## Research interests

My main research interests are directed towards the theory of continuous and discrete integrable systems and their connections with wider areas of pure and applied mathematics and with physics. A starting point for many of my contributions has been the study of integrable lattice equations, i.e., partial difference equations on the space-time lattice which share many of the beautiful features and properties with the more well-known integrable PDEs like the famous Korteweg-de Vries (KdV) equation. The study of solutions of such parial difference equations lead to conections with many areas of mathematics: soliton theory, algebraic geometry, Hamiltonian mechanics and the theory of symmetries and conservation laws. In fact, special solutions (i.e. reductions) lead to integrable dynamical mappings, discrete Painleve equations, and discrete systems of Calogero-Moser type. Fundamentally, these systems can also be quantized, and lead to corresponding systems in a discrete version of quantum mechanics. Furthermore, the study of the Lagrangian formalism of such systems has recently led to new paradigms in the theory of variational calculus.

### Useful links

Personal webpage

Full publication list

Teaching activities

## Current postgraduate students

Shami Alsallami (2014)

Alexander Bullivant (2013)

Wei Fu (2014)

Steven Haworth (2012)

Steven King (2013)

## Publications

**Hietarinta J; Joshi N; Nijhoff FW ** *Discrete Systems and Integrability*. Cambridge University Press 2016

View abstract

**Delice N, Nijhoff FW, Yoo-Kong S** On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants *Journal of Physics A: Mathematical and Theoretical*, **48**, 2015

DOI:10.1088/1751-8113/48/3/035206

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**Nijhoff FW, Jennings P** On an elliptic extension of the Kadomtsev–Petviashvili equation *Journal of Physics A: Mathematical and Theoretical*, **47**, 2014

DOI:10.1088/1751-8113/47/5/055205

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**Atkinson J, Lobb SB, Nijhoff FW** An integrable multicomponent quad-equation and its Lagrangian formulation *Theoretical and Mathematical Physics*, **173**, 1644-1653, 2012

DOI:10.1007/s11232-012-0138-y

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**Zhang DJ, Zhao SL, Nijhoff FW** Direct Linearization of Extended Lattice BSQ Systems *Studies in Applied Mathematics*, **129**, 220-248, 2012

DOI:10.1111/j.1467-9590.2012.00552.x

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**Xenitidis P, Nijhoff F** Symmetries and conservation laws of lattice Boussinesq equations *Physics Letters, Section A: General, Atomic and Solid State Physics*, **376**, 2394-2401, 2012

DOI:10.1016/j.physleta.2012.06.004

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**Zhang BD-J, Zhao S-L, Nijhoff FW** Direct Linearization of Extended Lattice BSQ Systems *Studies in Applied Mathematics*, 2012

DOI:10.1111/j.1467-9590.2012.00552.x

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**Xenitidis P, Nijhoff FW, Lobb S** On the Lagrangian formulation of multidimensionally consistent systems *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, **467**, 3295-3317, 2011

DOI:10.1098/rspa.2011.0124

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**Yoo-Kong S, Nijhoff FW, Lobb S** Discrete-time Calogero-Moser system and Lagrangian 1-form structure *Journal of Physics A:Math. Theor.*, **44**, 2011

DOI:10.1088/1751-8113/44/36/365203

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**Spicer PE, Nijhoff FW, van der Kamp PH** Higher analogues of the discrete-time Toda equation and the quotient-difference algorithm *NONLINEARITY*, **24**, 2229-2263, 2011

DOI:10.1088/0951-7715/24/8/006

**Atkinson J, Nijhoff FW** A constructive approach to the soliton solutions of integrable lattice equations *Communications in Mathematical Physics*, **299**, 283-304, 2010

DOI:10.1007/s00220-010-1076-x

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**Nijhoff FW, Atkinson J** Elliptic N-soliton solutions of ABS lattice equations *International Mathematics Research Notices*, **2010**, 3837-3895

DOI:10.1093/imrn/rnq010

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**Lobb SB, Nijhoff FW** Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy *J PHYS A-MATH THEOR*, **43**, 2010

DOI:10.1088/1751-8113/43/7/072003

**Lobb SB, Nijhoff FW, Quispel GRW** Lagrangian multiform structure for the lattice KP system *J PHYS A-MATH THEOR*, **42**, 2009

DOI:10.1088/1751-8113/42/47/472002

**Spicer PE, Nijhoff FW** Semi-classical Laguerre polynomials and a third-order discrete integrable equation *J PHYS A-MATH THEOR*, **42**, 2009

DOI:10.1088/1751-8113/42/45/454019

**Nijhoff FW, Atkinson J, Hietarinta J** Soliton Solutions for ABS Lattice Equations: I Cauchy Matrix Approach *Journal of Physics A:Math. Theor.*, **42**, 2009

DOI:10.1088/1751-8113/42/40/404005

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**Lobb S, Nijhoff FW** Lagrangian multiforms and multidimensional consistency *Journal of Physics A:Math. Theor.*, **42**, 2009

DOI:10.1088/1751-8113/42/45/454013

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**Field CM, Joshi N, Nijhoff FW** q-difference equations of KdV type and Chazy-type second-degree difference equations *J PHYS A-MATH THEOR*, **41**, 2008

DOI:10.1088/1751-8113/41/33/332005

**Atkinson J, Nijhoff FW** Solutions of Adler's lattice equation associated with 2-cycles of the Bäcklund transformation *Journal of Nonlinear Mathematical Physics*, **15**, 34-42, 2008

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**Atkinson J, Hietarinta J, Nijhoff FW** Seed and soliton solutions for Adler's lattice equation *Journal of Physics A: Mathematical and General*, **40**, 2007

DOI:10.1088/1751-8113/40/1/F01

**Atkinson J, Hietarinta J, Nijhoff FW** Seed and soliton solutions for Adler's lattice equation *Journal of Physics A : Mathematical and Theoretical*, **40**, F1-F8, 2007

DOI:10.1088/1751-8113/40/1/F01

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**Clarkson PA, Joshi N, Mazzocco M, Nijhoff FW, Noumi M** One hundred years of PVI, the Fuchs-Painleve equation - Preface *J PHYS A-MATH GEN*, **39**, 2006

**Tongas A, Nijhoff FW** A discrete Garnier type system from symmetry reduction on the lattice *Journal of Physics A: Mathematical and General*, **39**, 12191-12202, 2006

DOI:10.1088/0305-4470/39/39/S12

**Clarkson PA; Joshi N; Mazzocco M; Nijhoff FW; Noumi M ** *Special Issue celebrating one hundred years of PVI, the Fuchs-Painleve equation*. IOPP 2006

**Field CM, Nijhoff FW** Time-sliced path integrals with stationary states *J PHYS A-MATH GEN*, **39**, L309-L314, 2006

DOI:10.1088/0305-4470/39/20/L03

**Puttock SE; Nijhoff FW ** *A two-parameter elliptic extension of the lattice KdV system*Bilinear Integrable Systems: From Classical to Quatum, Continuous to Discrete, 245-251 2006

**Field CM, Nijhoff FW, Capel HW** Exact solutions of quantum mappings from the lattice KdV as multi-dimensional operator difference equations *J PHYS A-MATH GEN*, **38**, 9503-9527, 2005

DOI:10.1088/0305-4470/38/43/007

**Clarkson PA, Joshi N, Mazzocco M, Nijhoff FW, Noumi M** Special issue: One hundred years of PVI, the Fuchs-Painleve equation *J PHYS A-MATH GEN*, **38**, 2005

**Tongas AS, Nijhoff FW** The Boussinesq integrable system. Compatible lattice and continuum structures *Glasgow Mathematical Journal*, **47A**, 205-219, 2005

DOI:10.1017/S0017089505002417

**Tongas AS, Nijhoff FW** Generalised hyperbolic Ernst equations for an Einstein-Maxwell-Weyl Field *Journal of Physics A: Mathematical and General*, **38**, 895-906, 2005

DOI:10.1088/0305-4470/38/4/009

**Field CM, Nijhoff FW** Quantum discrete Dubrovin equations *Journal of Physics A: Mathematical and General*, **37**, 8065-8087, 2004

DOI:10.1088/0305-4470/37/33/007

**Joshi N, Nijhoff FW, Ormerod C** Lax pairs for ultra-discrete Painlev\'e cellular automata *Journal of Physics A: Mathematical and General*, **37**, L559-L565, 2004

DOI:10.1088/0305-4470/37/44/L03

**Field CM, Nijhoff FW** A note on modified Hamiltonians for numerical integrations admitting an exact invariant *Nonlinearity*, **16**, 1673-1683, 2003

DOI:10.1088/0951-7715/16/5/308

**Nijhoff FW, Puttock SE** On a Two-Parameter Extension of the Lattice KdV System Associated with an Elliptic Curve *Journal of Nonlinear Mathematical Physics*, **10**, 107-123, 2003

**Nijhoff FW** Lax pair for the Adler (lattice Krichever-Novikov) system *PHYS LETT A*, **297**, 49-58, 2002

**Hietarinta J, Nijhoff FW, Satsuma J** Symmetries and integrability of difference equations - Special issue dedicated to the subject of the SIDE IV meeting, held in Tokyo, 27 November 1 December 2000 - Preface *J PHYS A-MATH GEN*, **34**, 10337-10340, 2001

**Nijhoff FW, Walker AJ** The discrete and continuous Painleve VI hierarchy and the Garnier systems *Glasgow Mathematical Journal*, **43A**, 109-123, 2001

DOI:10.1017/S0017089501000106

**Nijhoff FW, Ramani A, Grammaticos B, Ohta Y** On discrete Painleve equations associated with the lattice KdV systems and the Painleve VI equation *STUD APPL MATH*, **106**, 261-314, 2001

**Kuznetsov VB; Nijhoff FW ** *Kowalevski Workshop on Mathematical Methods of Regular Dynamics*34 2001

**Nijhoff FW, Hietarinta J, Satsuma J** Special issue: Symmetries and Integrability of Difference Equations (SIDE IV) *Journal of Physics A: Mathematical and General*, **34**, 10337-10744, 2001

DOI:10.1088/0305-4470/34/48/001

**Nijhoff FW, Kuznetsov VB** Special Issue: Papers based on the Kowalevski Workshop on Mathematical Methods of Regular Dynamics *Journal of Physics A: Mathematical and General*, **34**, 2071-2073, 2001

DOI:10.1088/0305-4470/34/11/001

**Nijhoff FW, Hone A, Joshi N** On a Schwarzian PDE Associated with the KdV Hierarchy *Physics Letters. Section A: General, Atomic and Solid State Physics*, **A267**, 147-156, 2000

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**Nijhoff FW, Joshi N, Hone A** On the Discrete and Continuous Miura Chain Associated with the Sixth Painleve Equation *Physics Letters. Section A: General, Atomic and Solid State Physics*, **A264**, 396-406, 2000

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**Clarkson PA; Nijhoff FW ** *Symmetries and Integrability of Difference Equations*. Cambridge University Press 1999

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**Nijhoff FW, Chalykh OA** Bispectral Rings of Difference Operators *Russian Mathematical Surveys*, **54**, 644-645, 1999

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**Kuznetsov VB, Nijhoff FW, Sklyanin EK** Separation of Variables for the Ruijsenaars System *Communications in Mathematical Physics*, **189**, 855-877, 1997

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**Hietarinta J, Nijhoff FW** The Eight Tetrahedron Equations *Journal of Mathematical Physics*, **37**, 3603-3615, 1997

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**Nijhoff FW, Ragnisco D, Kuznetsov VB** Integrable Time-Discretization of the Ruijsenaars-Schneider Model *Communications in Mathematical Physics*, **176**, 681-700, 1996

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**Nijhoff FW, Delice N** On elliptic Lax pairs and isomonodromic deformation systems for
elliptic lattice equations *Advanced Studies in Pure Mathematics*

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