# School of Mathematics

## Prof Alexander V Mikhailov

Professor of Mathematical Physics

Applied Mathematics

### Contact details

Room: 9.305

Tel: +44 (0)113 3435176

Email: A.V.Mikhailov @ leeds.ac.uk

### Keywords

Classification of integrable systems

Integrability of differential and difference equations

Algebraic theory of differential and difference equations

Classification of Lax oprators and reduction problem

Automorphic and quasi-graded Lie algebras

Invers spectral transform

Soliton theory

Multi-scaling asymptotics

### Useful links

Integrable Systems Group

My homepage

Publications

## Current postgraduate students

George Berkeley (2012)

## Publications

**Mikhailov AV, Papamikos G, Wang JP** Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere *Letters in Mathematical Physics*, **106**, 973-996, 2016

DOI:10.1007/s11005-016-0855-5

View abstract

**Konstantinou-Rizos S, Mikhailov AV** Anticommutative extension of the Adler map *Journal of Physics A: Mathematical and Theoretical*, **49**, 1-7, 2016

DOI:10.1088/1751-8113/49/30/30LT03

View abstract

**Mikhailov AV, Papamikos G, Wang JP** Dressing method for the vector sine-Gordon equation and its soliton interactions *Physica D: Nonlinear Phenomena*, **325**, 53-62, 2016

DOI:10.1016/j.physd.2016.01.010

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**Grahovski GG, Konstantinou-Rizos S, Mikhailov AV** Grassmann extensions of Yang-Baxter maps *Journal of Physics A: Mathematical and Theoretical*, **49**, 2016

DOI:10.1088/1751-8113/49/14/145202

View abstract

**Mikhailov AV, Papamikos G, Wang JP** Darboux transformation with dihedral reduction group *JOURNAL OF MATHEMATICAL PHYSICS*, **55**, 2014

DOI:10.1063/1.4901224

**Garifullin RN, Mikhailov AV, Yamilov RI** Discrete equation on a square lattice with a nonstandard structure of generalized symmetries *Theoretical and Mathematical Physics*, **180**, 765-780, 2014

DOI:10.1007/s11232-014-0178-6

View abstract

**Konstantinou-Rizos S, Mikhailov AV** Yang-Baxter maps and finite reduction groups with degenerated orbits *Journal of Physics A: Mathematical and Theoretical*, 2014

**Grahovski GG, Mikhailov AV** Integrable discretisations for a class of nonlinear Schrodinger equations on Grassmann algebras *Physics Letters A*, **377**, 3254-3259, 2013

DOI:10.1016/j.physleta.2013.10.018

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**Khanizadeh F, Mikhailov AV, Wang JP** Darboux transformations and Recursion operators for differential--difference equations *Theoretical and Mathematical Physics*, **177**, 1606-1654, 2013

DOI:10.1007/s11232-013-0124-z

View abstract

**Mikhailov AV; Xenitidis P ** Second order integrability conditions for difference equations: an integrable equation. 2013

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**Konstantinou-Rizos S, Mikhailov A** Darboux transformations, finite reduction groups and related Yang-Baxter
maps *Journal of Physics A: Mathematical and Theoretical*, **46**, 2012

DOI:10.1088/1751-8113/46/42/425201

View abstract

**Gerdjikov V, Grahovski G, Valchev T, Mikhailov A** On Soliton Interactions for the Hierarchy of a Generalised Heisenberg Ferromagnetic Model on Su(3)/S(U(1)× U(2)) Symmetric Space *Journal of Geometry and Symmetry in Physics*, **25**, 23-55, 2012

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**Mikhailov AV, Wang JP** A new recursion operator for Adler’s equation in the Viallet form *Physics Letters, Section A: General, Atomic and Solid State Physics*, **375**, 3960-3963, 2011

DOI:10.1016/j.physleta.2011.09.018

View abstract

**Mikhailov AV, Wang JP, Xenitidis P** Cosymmetries and Nijenhuis recursion operators for difference equations *Nonlinearity*, **24**, 2079-2097, 2011

DOI:10.1088/0951-7715/24/7/009

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**Gerdjikov VS, Grahovski GG, Mikhailov AV, Valchev TI** RATIONAL BUNDLES AND RECURSION OPERATORS FOR INTEGRABLE EQUATIONS ON A.III-TYPE SYMMETRIC SPACES *THEOR MATH PHYS+*, **167**, 740-750, 2011

**Mikhailov AV, Wang JP, Xenitidis P** RECURSION OPERATORS, CONSERVATION LAWS, AND INTEGRABILITY CONDITIONS FOR DIFFERENCE EQUATIONS *THEOR MATH PHYS+*, **167**, 421-443, 2011

**Gerdjikov VS, Grahovski GG, Mikhailov AV, Valchev TI** Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces *SYMMETRY INTEGR GEOM*, **7**, 2011

DOI:10.3842/SIGMA.2011.096

**Gerdjikov VS, Mikhailov AV, Valchev TI** Recursion operators and reductions of integrable equations on symmetric spaces *Journal of Geometry and Symmetry in Physics*, **20**, 1-34, 2010

View abstract

**Gerdjikov VS, Mikhailov AV, Valchev TI** Reductions of integrable equations on A.III-type symmetric spaces *Journal of Physics A: Mathematical and Theoretical*, **43**, 2010

DOI:10.1088/1751-8113/43/43/434015

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Bhatt R; Mikhailov AV (2010) On the inconsistency of the Camassa-Holm model with the shallow water
theory.

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**Gurevich AV, Kuznetsov EA, Mesyats GA, Mikhailov AV, Musher SL, Novikov SP, Rubenchik AM, Sagdeev RZ, Smirnov GI, Sinai YG, Skrinsky AN, Fortov VE** Vladimir Evgen'evich Zakharov (on his 70th birthday) *PHYS-USP+*, **52**, 863-864, 2009

DOI:10.3367/UFNe.0179.200908n.0917

**Mikhailov AV ** *Integrability*. Berlin, Heidelberg: Springer 2009

DOI:10.1007/978-3-540-88111-7

View abstract

**Mikhailov A, Sokolov V** Symmetries of Differential Equations and the problem of Integrability In *Integrability*, 19-88, 2009

DOI:10.1007/978-3-540-88111-7

**Mikhailov AV, Novikov VS, Wang JP** Symbolic representation and classification of integrable systems In *Algebraic Theory of Differential Equations*, 156-216, 2009

**MacCallum MAH; Mikhailov AV ** *Algebraic Theory of Differential Equations*. Cambridge: Cambridge University Press 2009

**Mikhailov A** Introduction In *Integrability*, 1-18, 2009

DOI:10.1007/978-3-540-88111-7

**Mikhailov AV, Novikov VS, Wang JP** On classification of integrable nonevolutionary equations *Studies in Applied Mathematics*, **118**, 419-457, 2007

DOI:10.1111/j.1467-9590.2007.00376.x

View abstract

**Lombardo S, Mikhailov AV** Reduction groups and automorphic Lie algebras *Communications in Mathematical Physics*, **258**, 179-202, 2005

DOI:10.1007/s00220-005-1334-5

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**Mikhailov AV, Novikov VS, Wang JP** Partially integrable nonlinear equations with one higher symmetry *Journal of Physics A: Mathematical and General*, **38**, L337-L341, 2005

DOI:10.1088/0305-4470/38/20/L02

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**Lombardo S; Mikhailov AV ** *Reductions of integrable equations and automorphic Lie algebras*Conference on Symmetry and Perturbation Theory, 183-192 2005

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**Lombardo S, Mikhailov AV** Reductions of integrable equations: dihedral group *Journal of Physics A: Mathematical and General*, **37**, 7727-7742, 2004

DOI:10.1088/0305-4470/37/31/006

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**Mikhailov AV; Novokshenov VY ** *The Riemann-Hilbert problem for analytic description of the DM solitons*THEOR MATH PHYS+, 1723-1732 2003

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**Mikhailov AV, Novikov VS** Classification of Integrable Benjamin—Ono-Type Equations *Moscow Mathematical Journal*, **3**, 1293-1305, 2003

**Mikhailov AV, Novokshenov VY** Period map for nonlinear pulses in optical dispersion-managed fibers *THEOR MATH PHYS+*, **134**, 107-116, 2003

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**Mikhailov AV, Santini PM** Nonlinear Evolution Equations and Dynamical Systems *Theoretical and Mathematical Physics*, **134**, 1-40, 2003

DOI:10.1023/A:1021875924760

**Mikhailov AV, Novikov VS** Perturbative symmetry approach *Journal of Physics A: Mathematical and General*, **35**, 4775-4790, 2002

DOI:10.1088/0305-4470/35/22/309

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**Mikhailov AV, Santini PM** Nonlinear Evolution Equations and Dynamical Systems *Theoretical and Mathematical Physics*, **133**, 1607-1756, 2002

DOI:10.1023/A:1021396022516

**Kodama Y, Mikhailov AV** Symmetry and perturbation of the vector nonlinear Schrodinger equation *PHYSICA D*, **152**, 171-177, 2001

**Turitsyn SK, Mikhailov AV** Application of soloitons In *Scattering*, 1741-1753, 2001

**Mikhailov AV, Novokshenov VY** The period map for pulse propagation in nonlinear optical DM fibers *JETP LETT+*, **73**, 250-254, 2001

**Mikhailov AV, Sokolov VV** Integrable ODEs on Associative Algebras *Communications in Mathematical Physics*, **211**, 231-251, 2000

DOI:10.1007/s002200050810

View abstract

**Leon J, Mikhailov AV** Raman Soliton Generation From Laser Inputs in SRS *Physics Letters. Section A: General, Atomic and Solid State Physics*, **253**, 33-40, 1999

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**Galeev AA; Kuznetsov EA; Mikhailov AV; Nesterikhin YE; Novikov SP; Sagdeev RZ; Sinai YG; Smirnov GI; Fortov VE; Khalatnikov IM ** Vladimir Evgenevich Zakharov (on his sixtieth birthday). **169**, 1161-1162 1999

**Zakharov VE ed. ** *Transient Raman amplification* 1999

**Zakharov VEWS ed. ** *Variationalism and empirio-criticism. (Exact and Variational Approaches to fibre optics equations)* 1999

**Mikhailov AV, Yamilov RI** Towards classification of (2 + 1)-dimensional integrable equations. Integrability conditions I *Journal of Physics A: Mathematical and General*, **31**, 6707-6715, 1998

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**Leon J; Mikhailov AV ** Raman Solitons and Raman spikes. 1997

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**Kodama Y, Mikhailov AV, Wabnitz S** Input Pulse Optimization in Wavelength-Division-Multiplexed Soliton Transmissions *Optics Communications*, **143**, 53-56, 1997

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**Montes C, Mikhailov A, Picozzi A, Ginovart F** Dissipative three-wave structures in stimulated backscattering .1. A subluminous solitary attractor *PHYSICAL REVIEW E*, **55**, 1086-1091, 1997

DOI:10.1103/PhysRevE.55.1086

**Mikhailov AV; Yamilov RI ** On integrable two-dimensional generalizations of nonlinear Schrodinger type equations. **230**, 295-300 1997

**Mikhailov AV; Kuznetsov EA; Newell AC; Zakharov VE ** THE NONLINEAR SCHRODINGER-EQUATION - PROCEEDINGS OF THE CONFERENCE ON THE NONLINEAR SCHRODINGER-EQUATION - CHERNOGOLOVKA, 25 JULY 3 AUGUST 1994. **87**, R7-R9 1995

**Kuznetsov EA; Mikhailov AV; Shimokhin IA ** NONLINEAR-INTERACTION OF SOLITONS AND RADIATION. **87**, 201-215 1995

**Kuznetsov EA; Mikhailov AV ** RELAXATION OSCILLATIONS OF SOLITONS. **60**, 486-490 1994

**Devega HJ; Mikhailov AV; Sanchez N ** EXACT STRING SOLUTIONS IN (2+1)-DIMENSIONAL DESITTER SPACE-TIME. **9**, 2745-2754 1994

**Devega HJ, Mikhailov AV, Sanchez N** NEW SOLUTIONS OF THE CLASSICAL STRING EQUATION IN DESITTER SPACE In *Nonlinear Coherent Structures in Physics and Biology*, 271-274, 1994

**Combes F, Vega HJD, Mikhailov AV, Sánchez N** Multi-String Solutions by Soliton Methods in De Sitter Spacetime *Physical Review D*, **50**, 2754-2768, 1993

DOI:10.1103/PhysRevD.50.2754

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**Vega HJD, Mikhailov AV, Sanchez N** Exact String Solutions in 2+1-Dimensional De Sitter Spacetime *Theoretical and Mathematical Physics*, **94**, 166-172, 1993

DOI:10.1007/BF01019328

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**Mikhailov AV; Shimokhin IA ** SPECTRUM OF 180-DEGREES BLOCH-TYPE DOMAIN-WALL EXCITATIONS IN YTTRIUM-IRON-GARNET. **48**, 9569-9574 1993

**Mikhailov AV; Wabnitz S ** POLARIZATION DYNAMICS OF COUNTERPROPAGATING BEAMS IN OPTICAL FIBERS. **15**, 1055-1057 1990

**Conte RBN ed. ** *A CONCEPT OF INTEGRABILITY BASED ON THE SYMMETRY APPROACH*. 310th ed 1990

**Mikhailov AV; Shimokhin IA ** DOMAIN-WALL EXCITATION SPECTRUM IN UNIAXIAL FERROMAGNET. **97**, 1966-1973 1990

**Mikhailov AV; Shabat AB; Yamilov RI ** EXTENSION OF THE MODULE OF INVERTIBLE TRANSFORMATIONS - CLASSIFICATION OF INTEGRABLE SYSTEMS. **115**, 1-19 1988

**Mikhailov AV; Shabat AB; Iamilov RI ** ON THE EXTENSION OF REVERSIBLE TRANSFORMATION MODULE. **295**, 288-291 1987

**Mikhailov AV; Shabat AB; Yamilov RI ** THE SYMMETRY APPROACH TO THE CLASSIFICATION OF NON-LINEAR EQUATIONS - COMPLETE LISTS OF INTEGRABLE SYSTEMS. **42**, 1-63 1987

**Burtsev SP; Zakharov VE; Mikhailov AV ** INVERSE SCATTERING METHOD WITH VARIABLE SPECTRAL PARAMETER. **70**, 227-240 1987

**Zakharov VE; Mikhailov AV ** POLARIZATION DOMAINS IN NONLINEAR OPTICS. **45**, 349-352 1987

**Mikhailov AV; Shabat AB ** INTEGRABILITY CONDITIONS FOR SYSTEMS OF 2 EQUATIONS OF THE FORM UT = A(U)UXX + F(U, UX) .2. **66**, 31-44 1986

**Mikhailov AV; Shabat AB ** INTEGRABLE DEFORMATIONS OF THE HEISENBERG-MODEL. **116**, 191-194 1986

**Mikhailov AV; Shabat AB ** INTEGRABILITY CONDITIONS FOR SYSTEMS OF 2 EQUATIONS OF THE FORM UT = A(U)UXX + F(U,UX) .1. **62**, 107-122 1985

**Gabitov IR; Zakharov VE; Mikhailov AV ** MAXWELL-BLOCH EQUATION AND THE INVERSE SCATTERING METHOD. **63**, 328-343 1985

**Gabitov IR; Zakharov VE; Mikhailov AV ** NONLINEAR-THEORY OF SUPERFLUORESCENCE. **86**, 1204-1216 1984

**Mikhailov AV; Yaremchuk AI ** FORCED MOTION OF A DOMAIN-WALL IN THE FIELD OF SPIN-WAVE. **39**, 354-357 1984

**Zakharov VE; Mikhailov AV ** METHOD OF THE INVERSE SCATTERING PROBLEM WITH SPECTRAL PARAMETER ON AN ALGEBRAIC CURVE. **17**, 247-251 1983

**Gabitov IR; Zakharov VE; Mikhailov AV ** SUPERFLUORESCENCE PULSE SHAPE. **37**, 279-282 1983

**Mikhailov AV ** RELATIVISTICALLY INVARIANT MODELS OF THE FIELD-THEORY INTEGRABLE BY THE INVERSE SCATTERING METHOD. **970**, 186-196 1982

**Mikhailov AV ** THE LANDAU-LIFSCHITZ EQUATION AND THE RIEMANN BOUNDARY-PROBLEM ON A TORUS. **92**, 51-55 1982

**Mikhailov AV; Yaremchuk AI ** AXIALLY SYMMETRICAL SOLUTIONS OF THE TWO-DIMENSIONAL HEISENBERG-MODEL. **36**, 95-100 1982

**Mikhailov AV; Yaremchuk AI ** CYLINDRICALLY SYMMETRIC-SOLUTIONS OF THE NON-LINEAR CHIRAL FIELD MODEL (SIGMA-MODEL). **202**, 508-522 1982

**Mikhailov AV ** THE REDUCTION PROBLEM AND THE INVERSE SCATTERING METHOD. **3**, 73-117 1981

**Mikhailov AV; Olshanetsky MA; Perelomov AM ** TWO-DIMENSIONAL GENERALIZED TODA LATTICE. **79**, 473-488 1981

**Lvov VS; Mikhailov AV ** CONTRIBUTION TO THE NON-LINEAR THEORY OF SOUND AND HYDRODYNAMIC TURBULENCE OF A COMPRESSIBLE LIQUID. **2**, 224-243 1981

**Kuznetsov EA; Mikhailov AV ** ON THE TOPOLOGICAL MEANING OF CANONICAL CLEBSCH VARIABLES. **77**, 37-38 1980

**Mikhailov AV ** REDUCTION IN INTEGRABLE SYSTEMS - THE REDUCTION GROUP. **32**, 174-178 1980

**Zakharov VE; Mikhailov AV ** ON THE INTEGRABILITY OF CLASSICAL SPINOR MODELS IN TWO-DIMENSIONAL SPACE-TIME. **74**, 21-40 1980

**Mikhailov AV; Perelomov AM ** INSTANTON-TYPE SOLUTIONS IN SUPER-SYMMETRICAL CHIRAL MODELS. **29**, 403-407 1979

**Mikhailov AV ** INTEGRABILITY OF A TWO-DIMENSIONAL GENERALIZATION OF THE TODA CHAIN. **30**, 414-418 1979

**Mikhailov AV ** INTEGRABILITY OF SUPERSYMMETRICAL GENERALIZATIONS OF CLASSICAL CHIRAL MODELS IN 2-DIMENSIONAL SPACE-TIME. **28**, 512-516 1978

**Lvov VS; Mikhailov AV ** SCATTERING AND INTERACTION OF SOUND WITH SOUND IN A TURBULENT MEDIUM. **75**, 1669-1682 1978

**Lvov VS; Mikhailov AV ** SOUND AND HYDRODYNAMIC TURBULENCE IN A COMPRESSIBLE LIQUID. **74**, 1445-1457 1978

**Zakharov VE; Mikhailov AV ** EXAMPLE OF NONTRIVIAL INTERACTION OF SOLITONS IN 2-DIMENSIONAL CLASSICAL FIELD-THEORY. **27**, 42-46 1978

**Zakharov VE; Mikhailov AV ** RELATIVISTICALLY INVARIANT 2-DIMENSIONAL MODELS IN FIELD-THEORY INTEGRABLE BY INVERSE PROBLEM TECHNIQUE. **74**, 1953-1973 1978

**Kuznetsov EA; Mikhailov AV ** COMPLETE INTEGRABILITY OF 2-DIMENSIONAL CLASSICAL THIRRING MODEL. **30**, 193-200 1977

**Mikhailov AV ** INTEGRABILITY OF 2-DIMENSIONAL THIRRING MODEL. **23**, 320-323 1976

**Kuznetso.Ea; Mikhailo.Av ** STABILITY OF STATIONARY WAVES IN NONLINEAR WEAKLY DISPERSIVE MEDIA. **67**, 1717-1727 1974

**Berkeley G, Mikhailov AV, Xenitidis P** Darboux transformations with tetrahedral reduction group and related integrable systems *Journal of Mathematical Physics*

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