# School of Mathematics

## Prof Allan P Fordy

Professor of Nonlinear Dynamics

Applied Mathematics

### Contact details

Room: 8.23

Tel: +44 (0)113 3435115

Email: A.P.Fordy @ leeds.ac.uk

### Keywords

Integrable Systems

Completely Integrable Hamiltonian Systems

Mathematical Physics

Cluster Algebras

Integrable Maps

Superintegrability and Poisson Algebras

## Research interests

My main research interests lie within Integrable Systems. Most of my research has been of an algebraic nature: the classification of Lax pairs through Lie algebras and symmetric spaces; Hamiltonian structures and Miura maps; stationary flows and the reduction of Poisson brackets; symmetries and superintegrability in quantum integrable systems; Poisson algebras related to classical superintegrable systems; integrable maps related to cluster algebras and associated Poisson algebras.

The main thrust of my recent research has been in various aspects of integrable maps.

In particular we recently introduced a new class of mutation-periodic quivers and gave a full classification of the period one case. Later papers are analysing and classifying the associated integrable maps.

### Useful links

Integrable Systems Group

Personal Home Page

Google Scholar Research Profile

## Publications

**Fordy AP, Xenitidis P** Self-dual systems, their symmetries and reductions to the Bogoyavlensky lattice *Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)*, **13**, 2017

DOI:10.3842/SIGMA.2017.051

View abstract

**Fordy AP, Xenitidis P** ZN graded discrete Lax pairs and Yang–Baxter maps *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science*, **473**, 20160946-20160946, 2017

DOI:10.1098/rspa.2016.0946

**Fordy AP** A note on some superintegrable Hamiltonian systems *Journal of Geometry and Physics*, **115**, 98-103, 2017

DOI:10.1016/j.geomphys.2016.06.005

**Fordy AP, Xenitidis P** ${{\mathbb{Z}}_{N}}$ graded discrete Lax pairs and integrable difference equations *Journal of Physics A: Mathematical and Theoretical*, **50**, 165205-165205, 2017

DOI:10.1088/1751-8121/aa639a

**Fordy AP** Periodic cluster mutations and related integrable maps *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, **47**, 2014

DOI:10.1088/1751-8113/47/47/474003

**Fordy AP, Hone A** Discrete Integrable Systems and Poisson Algebras From Cluster Maps *COMMUNICATIONS IN MATHEMATICAL PHYSICS*, **325**, 527-584, 2014

DOI:10.1007/s00220-013-1867-y

**Fordy AP, Kassotakis P** Integrable maps which preserve functions with symmetries *Journal of Physics A: Mathematical and Theoretical*, **46**, 2013

DOI:10.1088/1751-8113/46/20/205201

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**Fordy AP, Scott MJ** Recursive procedures for Krall-Sheffer operators *Journal of Mathematical Physics*, **54**, 2013

DOI:10.1063/1.4802098

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**Fordy AP, Marsh RJ** Cluster mutation-periodic quivers and associated Laurent sequences *J ALGEBR COMB*, **34**, 19-66, 2011

DOI:10.1007/s10801-010-0262-4

**Fordy AP** Mutation-periodic quivers, integrable maps and associated Poisson algebras *PHILOS T R SOC A*, **369**, 1264-1279, 2011

DOI:10.1098/rsta.2010.0318

**Fordy AP, Hone A** Symplectic Maps from Cluster Algebras *Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)*, **7**, 2011

DOI:10.3842/SIGMA.2011.091

View abstract

**Fordy AP** Quantum Super-Integrable Systems as Exactly Solvable Models *SYMMETRY INTEGR GEOM*, **3**, 2007

**Fordy AP** Darboux related quantum integrable systems on a constant curvature surface *Journal of Geometry and Physics*, **56**, 1709-1727, 2006

DOI:10.1016/j.geomphys.2005.10.001

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**Fordy AP, Kassotakis PG** Multidimensional maps of QRT type *J PHYS A-MATH GEN*, **39**, 10773-10786, 2006

DOI:10.1088/0305-4470/39/34/012

**Fordy AP** Symmetries, Ladder Operators and Quantum Integrable systems *Glasgow Mathematical Journal*, **47A**, 65-75, 2005

DOI:10.1017/S0017089505002296

View abstract

**Fordy AP, Mokhov OI** On a special class of compatible Poisson structures of hydrodynamic type *Physica D - Nonlinear Phenomena*, **152-153**, 475-490, 2001

DOI:10.1016/S0167-2789(01)00187-7

View abstract

**Fordy AP, Gurel TB** A New Construction of Recursion Operators for Systems of Hydrodynamic Type *Theoretical and Mathematical Physics*, **122**, 29-38, 2000

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**Fordy AP ** *Systems of hydrodynamic type from Poisson commuting Hamiltonians*Meeting on Integrability - The Seiberg-Witten and Whitham Equations, 229-249 2000

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**Ferapontov EV, Fordy AP** Commuting Quadratic Hamiltonians with Velocity Dependent Potentials *Reports on Mathematical Physics*, **44**, 71-80, 1999

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**Ferapontov EV, Fordy AP** Non-homogeneous systems of hydrodynamic type, related to quadratic Hamiltonians with electromagnetic term *Physica D: Nonlinear Phenomena*, **108**, 350-364, 1997

DOI:10.1016/S0167-2789(97)00040-7

**Fordy AP, Harris SD** Hamiltonian Flows on Stationary Manifolds *Methods and Applications of Analysis*, **4**, 212-238, 1997

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**Ferapontov EV, Fordy AP** Separable Hamiltonians and Integrable Systems of Hydrodynamic Type *Journal of Geometry and Physics*, **21**, 169-182, 1997

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**Fordy AP, Shabat AB, Veselov AP** Factorization and Poisson correspondences *Theoretical and Mathematical Physics*, **105**, 1369-1386, 1995

DOI:10.1007/BF02070933

**Fordy AP** Stationary flows: Hamiltonian structures and canonical transformations *Physica D: Nonlinear Phenomena*, **87**, 20-31, 1995

DOI:10.1016/0167-2789(95)00134-P

**Baker S, Enolskii VZ, Fordy AP** Integrable quartic potentials and coupled KdV equations *Physics Letters A*, **201**, 167-174, 1995

DOI:10.1016/0375-9601(95)00267-7

**Conte R, Fordy AP, Pickering A** A perturbative Painlevé approach to nonlinear differential equations *Physica D: Nonlinear Phenomena*, **69**, 33-58, 1993

DOI:10.1016/0167-2789(93)90179-5

**Fordy AP, Holm DD** A tri-Hamiltonian formulation of the self-induced transparency equations *Physics Letters A*, **160**, 143-148, 1991

DOI:10.1016/0375-9601(91)90603-6

**Fordy A, Pickering A** Analysing negative resonances in the Painlevé test *Physics Letters A*, **160**, 347-354, 1991

DOI:10.1016/0375-9601(91)90663-S

**Fordy AP** The Hénon-Heiles system revisited *Physica D: Nonlinear Phenomena*, **52**, 204-210, 1991

DOI:10.1016/0167-2789(91)90122-P

**Antonowicz M, Fordy AP, Liu QP** Energy-dependent third-order Lax operators *Nonlinearity*, **4**, 669-684, 1991

DOI:10.1088/0951-7715/4/3/003

**Antonowicz M, Fordy AP** Factorisation of energy dependent Schr�dinger operators: Miura maps and modified systems *Communications in Mathematical Physics*, **124**, 465-486, 1989

DOI:10.1007/BF01219659

**Antonowicz M, Fordy AP** Super-extensions of energy dependent Schr�dinger operators *Communications in Mathematical Physics*, **124**, 487-500, 1989

DOI:10.1007/BF01219660

**Fordy AP, Reyman AG, Semenov-Tian-Shansky MA** Classical r-matrices and compatible Poisson brackets for coupled KdV systems *Letters in Mathematical Physics*, **17**, 25-29, 1989

DOI:10.1007/BF00420010

**Antonowicz M, Fordy AP** Coupled Harry Dym equations with multi-Hamiltonian structures *Journal of Physics A: Mathematical and General*, **21**, L269-L275, 1988

DOI:10.1088/0305-4470/21/5/001

**Antonowicz M, Fordy AP** Coupled KdV equations with multi-Hamiltonian structures *Physica D: Nonlinear Phenomena*, **28**, 345-357, 1987

DOI:10.1016/0167-2789(87)90023-6

**Fordy A, Athorne C** Generalized Davey-Stewartson equations associated with symmetric spaces *Physica D: Nonlinear Phenomena*, **28**, 248-248, 1987

DOI:10.1016/0167-2789(87)90206-5

**Fordy A** Some completely integrable classical mechanical systems associated with symmetric spaces *Physica D: Nonlinear Phenomena*, **28**, 242-242, 1987

DOI:10.1016/0167-2789(87)90191-6

**Antonowicz M, Fordy AP, Wojciechowski S** Integrable stationary flows: Miura maps and bi-hamiltonian structures *Physics Letters A*, **124**, 143-150, 1987

DOI:10.1016/0375-9601(87)90241-6

**Athorne C, Fordy A** Integrable equations in (2+1) dimensions associated with symmetric and homogeneous spaces *Journal of Mathematical Physics*, **28**, 2018-2024, 1987

DOI:10.1063/1.527463

**Antonowicz M, Fordy AP** A family of completely integrable multi-hamiltonian systems *Physics Letters A*, **122**, 95-99, 1987

DOI:10.1016/0375-9601(87)90783-3

**Athorne C, Fordy A** Generalised KdV and MKdV equations associated with symmetric spaces *Journal of Physics A: Mathematical and General*, **20**, 1377-1386, 1987

DOI:10.1088/0305-4470/20/6/021

**Fordy A, Wojciechoski S, Marshall I** A family of integrable quartic potentials related to symmetric spaces *Physics Letters A*, **113**, 395-400, 1986

DOI:10.1016/0375-9601(86)90658-4

**Dodd RK, Fordy AP** Prolongation structures of complex quasi-polynomial evolution equations *Journal of Physics A: Mathematical and General*, **17**, 3249-3266, 1984

DOI:10.1088/0305-4470/17/16/025

**Fordy AP** Derivative nonlinear Schrodinger equations and Hermitian symmetric spaces *Journal of Physics A: Mathematical and General*, **17**, 1235-1245, 1984

DOI:10.1088/0305-4470/17/6/019

**Fordy AP, Kulish PP** Nonlinear Schr�dinger equations and simple Lie algebras *Communications in Mathematical Physics*, **89**, 427-443, 1983

DOI:10.1007/BF01214664

**Fordy AP** Hamiltonian symmetries of the Henon-Heiles system *Physics Letters A*, **97**, 21-23, 1983

DOI:10.1016/0375-9601(83)90091-9

**Dodd R, Fordy A** The Prolongation Structures of Quasi-Polynomial Flows *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, **385**, 389-429, 1983

DOI:10.1098/rspa.1983.0020

**Bullough RK, Fordy AP, Manakov SV** Adiabatic invariants theory of near-integrable systems with damping *Physics Letters A*, **91**, 98-100, 1982

DOI:10.1016/0375-9601(82)90358-9

**Dodd R, Fordy A** On the integrability of a system of coupled KdV equations *Physics Letters A*, **89**, 168-170, 1982

DOI:10.1016/0375-9601(82)90199-2

**Fordy AP, Gibbons J** Factorization of operators.II *Journal of Mathematical Physics*, **22**, 1170-1175, 1981

DOI:10.1063/1.525041

**Fordy AP, Gibbons J** Integrable nonlinear Klein-Gordon equations and Toda lattices *Communications in Mathematical Physics*, **77**, 21-30, 1980

DOI:10.1007/BF01205037

**Fordy AP, Gibbons J** Some remarkable nonlinear transformations *Physics Letters A*, **75**, 325-325, 1980

DOI:10.1016/0375-9601(80)90829-4

**Fordy AP, Gibbons J** Factorization of operators I. Miura transformations *Journal of Mathematical Physics*, **21**, 2508-2508, 1980

DOI:10.1063/1.524357

**Fordy AP** Higher-spin fields in a curved space-time *General Relativity and Gravitation*, **9**, 387-392, 1978

DOI:10.1007/BF00759839

**Fordy AP** Zero-rest-mass fields in an algebraically special curved space-time *General Relativity and Gravitation*, **8**, 227-243, 1977

DOI:10.1007/BF00765810

Fordy AP; Huang Q Poisson Algebras and 3D Superintegrable Hamiltonian Systems.

View abstract

Fordy AP; Xenitidis P ${\mathbb{Z}}_N$ graded discrete Lax pairs and discrete integrable
systems.

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Fordy AP; Xenitidis P ${\mathbb{Z}}_N$ graded discrete Lax pairs and Yang-Baxter maps.

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