Portrait of Professor Peter A Clarkson

Professor Peter A Clarkson

Professor of Mathematics

About

Peter chairs the School's EDI Committee and is a member of the University's Athena SWAN working group. In addition to his work at the University Peter is a member of the London Mathematical Society's Women in Mathematics Committee and Chair of the steering committee for the LMS Good Practice Scheme. The Good Practice Scheme has the the aim of supporting Mathematics departments interested in embedding equal opportunities for women within their working practices. The Scheme provides specific support for departments working towards Athena SWAN Award status and organizes events.

Research interests

  • Soliton theory, in particular the Painlevé equations, and Painlevé analysis.
  • Asymptotics, Bäcklund transformations, connection formulae and exact solutions for nonlinear ordinary differential and difference equations, in particular the Painlevé equations.
  • Orthogonal polynomials and special functions, in particular nonlinear special functions such as the Painlevé equations.
  • Symmetry reductions and exact solutions of nonlinear partial differential equations, in particular using nonclassical and generalized techniques. equations.

Professional

Peter is a participant in the NIST Digital Library of Mathematical Functions project, companion to the NIST Handbook of Mathematical Functions, funded by the U.S. National Science Foundation, and organised by the National Institute of Standards and Technology, Gaithersburg, Maryland, USA. This project is to update Abramowitz and Stegun's Handbook of Mathematical Functions. Peter's role in the project is with writing the chapter on Painlevé Transcendents.  

Publications

Showing 50 of 76 total publications in the Kent Academic Repository. View all publications.

Article

  • Clarkson, P. (2019). Open Problems for Painlevé Equations. Symmetry, Integrability and Geometry: Methods and Applications [Online] 15:6. Available at: https://doi.org/10.3842/SIGMA.2019.006.
    In this paper some open problems for Painlevé equations are discussed. In
    particular the following open problems are described: (i) the Painlevé equivalence problem;
    (ii) notation for solutions of the Painlevé equations; (iii) numerical solution of Painlevé
    equations; and (iv) the classification of properties of Painlevé equations.
  • Rogers, C. and Clarkson, P. (2018). Ermakov-Painlevé II Reduction in Cold Plasma Physics. Application of a Bäcklund Transformation. Journal of Nonlinear Mathematical Physics [Online] 25:247-261. Available at: https://doi.org/10.1080/14029251.2018.1452672.
    A class of symmetry transformations of a type originally introduced in a nonlinear optics
    context is used here to isolate an integrable Ermakov-Painlev´e II reduction of a resonant NLS
    equation which encapsulates a nonlinear system in cold plasma physics descriptive of the uniaxial
    propagation of magneto-acoustic waves. A B¨acklund transformation is employed in the
    iterative generation of novel classes of solutions to the cold plasma system which involve either
    Yablonski-Vorob’ev polynomials or classical Airy functions
  • Rogers, C., Bassom, A. and Clarkson, P. (2018). On integrable Ermakov-Painlevé IV systems. Journal of Mathematical Analysis and Applications [Online] 462:1225-1241. Available at: https://doi.org/10.1016/j.jmaa.2018.02.025.
    Novel hybrid Ermakov-Painleve IV systems are introduced and an associated Ermakov invariant is used in establishing their integrability. Backlund transformations are then employed to generate classes of exact solutions via the linked canonical Painleve IV equation.
  • Rogers, C. and Clarkson, P. (2017). Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System. Symmetry, Integrability and Geometry: Methods and Applications [Online] 13. Available at: https://doi.org/10.3842/SIGMA.2017.018.
    A class of nonlinear Schrödinger equations involving a triad of power law terms together with a de Broglie-Bohm potential is shown to admit symmetry reduction to a hybrid Ermakov-Painlevé II equation which is linked, in turn, to the integrable Painlevé XXXIV equation. A nonlinear Schrödinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov-Painlevé II reduction valid for a multi-parameter class of free energy functions. Iterated application of a Bäcklund transformation then allows the construction of novel classes of exact solutions of the nonlinear capillarity system in terms of Yablonskii-Vorob'ev polynomials or classical Airy functions. A Painlevé XXXIV equation is derived for the density in the capillarity system and seen to correspond to the symmetry reduction of its Bernoulli integral of motion
  • Liu, T. et al. (2017). A connection between the maximum displacements of rogue waves and the dynamics of poles in the complex plane. Chaos: An Interdisciplinary Journal of Nonlinear Science [Online] 27:91103. Available at: http://dx.doi.org/10.1063/1.5001007.
    Rogue waves of evolution systems are displacements which are localized in both space and time. The locations of the points of maximum displacements of the wave profiles may correlate with the trajectories of the poles of the exact solutions from the perspective of complex variables through analytic continuation. More precisely, the location of the maximum height of the rogue wave in laboratory coordinates (real space and time) is conjectured to be equal to the real part of the pole of the exact solution, if the spatial coordinate is allowed to be complex. This feature can be verified readily for the Peregrine breather (lowest order rogue wave) of the nonlinear Schrödinger equation. This connection is further demonstrated numerically here for more complicated scenarios, namely the second order rogue wave of the Boussinesq equation (for bidirectional long waves in shallow water), an asymmetric second order rogue wave for the nonlinear Schrödinger equation (as evolution system for slowly varying wave packets), and a symmetric second order rogue wave of coupled Schrödinger systems. Furthermore, the maximum displacements in physical space occur at a time instant where the trajectories of the poles in the complex plane reverse directions. This property is conjectured to hold for many other systems, and will help to determine the maximum amplitudes of rogue waves.
  • Clarkson, P. and Jordaan, K. (2017). Properties of Generalized Freud Polynomials. Journal of Approximation Theory [Online] 225:148-175. Available at: https://doi.org/10.1016/j.jat.2017.10.001.
  • Clarkson, P. and Dowie, E. (2017). Rational solutions of the Boussinesq equation and applications to rogue waves. Transactions of Mathematics and Its Applications [Online] 1. Available at: https://doi.org/10.1093/imatrm/tnx003.
    We study rational solutions of the Boussinesq equation, which is a soliton equation solvable by the inverse scattering method. These rational solutions, which are algebraically decaying and depend on two arbitrary parameters, are expressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the focusing nonlinear Schr¨odinger (NLS) equation. Further the rational solutions have an interesting structure as they are comprised of a linear combination of four independent solutions of the bilinear equation. Rational solutions of the Kadomtsev-Petviashvili I (KPI) equation are derived in two ways, from rational solutions of the NLS equation and from rational solutions of the Boussinesq equation. It is shown that these two families of rational solutions of the KPI equation are fundamentally different and a unifying framework is found which incorporates both families of solutions.
  • Ankiewicz, A. et al. (2017). Conservation Laws and Integral Relations for the Boussinesq Equation. Studies in Applied Mathematics [Online]. Available at: https://dx.doi.org/10.1111/sapm.12174.
    We are concerned with conservation laws and integral relations associated with rational solutions of the Boussinesq equation, a soliton equation solvable by inverse scattering, which was first introduced by Boussinesq in 1871. The rational solutions are logarithmic derivatives of a polynomial, are algebraically decaying, and have a similar appearance to rogue-wave solutions of the focusing nonlinear Schrödinger equation. For these rational solutions, the constants of motion associated with the conserved quantities are zero and they have some interesting integral relations, which depend on the total degree of the associated polynomial.
  • Clarkson, P., Loureiro, A. and Van Assche, W. (2016). Unique positive solution for an alternative discrete Painlevé I equation. Journal of Difference Equations and Applications [Online]. Available at: http://www.tandfonline.com/doi/abs/10.1080/10236198.2015.1127917.
    We show that the alternative discrete Painleve I equation has a unique solution which remains positive for all n >0. Furthermore, we identify this positive solution in terms of a special solution of the second Painleve equation involving the Airy function Ai(t). The special-function solutions of the second Painleve equation involving only the Airy function Ai(t) therefore have the property that they remain positive for all n>0 and all t>0, which is a new characterization of these special solutions of the second Painlevé equation and the alternative discrete Painlevé I equation.
  • Clarkson, P. (2016). On Airy Solutions of the Second Painleve Equation. Studies in Applied Mathematics [Online] 137:93-109. Available at: http://dx.doi.org/10.1111/sapm.12123.
    In this paper, we discuss Airy solutions of the second Painleve equation and two related equations, the Painleve XXXIV equation and the Jimbo-Miwa-Okamoto sigma form of second Painleve equation, are discussed. It is shown that solutions which depend only on the Airy function Ai(z) have a completely difference structure to those which involve a linear combination of the Airy functions Ai(z) and Bi(z). For all three equations, the special solutions that depend only on inline image are tronquée solutions, i.e., they have no poles in a sector of the complex plane. Further, for both inline image and SII, it is shown that among these tronquée solutions there is a family of solutions that have no poles on the real axis.
  • Clarkson, P., Jordaan, K. and Kelil, A. (2015). A Generalized Freud Weight. Studies in Applied Mathematics [Online] 136:288-320. Available at: http://dx.doi.org/10.1111/sapm.12105.
    We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight
    w(x;t)=|x|2?+1exp(?x4+tx2),x?R,
    with parameters ?>?1 and t?R, and classical solutions of the fourth Painlev\'e equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlev\'e equation. Further we derive a second-order linear ordinary differential equation and a differential-difference equation satisfied by the generalized Freud polynomials.
  • Clarkson, P. and Jordaan, K. (2014). The Relationship Between Semiclassical Laguerre Polynomials and the Fourth Painlevé Equation. Constructive Approximation [Online] 39:223-254. Available at: http://dx.doi.org/10.1007/s00365-013-9220-4.
    We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation.
  • Clarkson, P. and Jordaan, K. (2014). The relationship between semi-classical Laguerre polynomials and the fourth Painlevé equation. Constructive Approximation [Online] 39:223-254. Available at: http://link.springer.com/article/10.1007/s00365-013-9220-4.
    We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painlevé equation
  • Clarkson, P. (2013). Recurrence coefficients for discrete orthonormal polynomials and the Painlevé equations. Journal of Physics A: Mathematical and Theoretical [Online] 46. Available at: http://dx.doi.org/10.1088/1751-8113/46/18/185205.
    We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlevé equations.
  • Ankiewicz, A., Clarkson, P. and Akhmediev, N. (2010). Rogue waves, rational solutions, the patterns of their zeros and integral relations. Journal of Physics A: Mathematical and Theoretical [Online] 43:122002. Available at: http://dx.doi.org/10.1088/1751-8113/43/12/122002.
    The focusing nonlinear Schrödinger equation, which describes generic nonlinear phenomena, including waves in the deep ocean and light pulses in optical fibres, supports a whole hierarchy of recently discovered rational solutions. We present recurrence relations for the hierarchy, the pattern of zeros for each solution and a set of integral relations which characterizes them.
  • Clarkson, P. (2009). Vortices and Polynomials. Studies in Applied Mathematics [Online] 123:37-62. Available at: http://dx.doi.org/10.1111/j.1467-9590.2009.00446.x.
    The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler–Moser polynomials, which arise in the description of rational solutions of the Korteweg–de Vries equation. For quadrupole background flow, vortex configurations are given by the zeros of polynomials expressed as Wronskians of Hermite polynomials. Further, new solutions are found in this case using the special polynomials arising in the description of rational solutions of the fourth Painlevé equation.
  • Clarkson, P. (2009). Rational solutions of the classical Boussinesq system. Nonlinear Analysis: Real World Applications [Online] 10:3360-3371. Available at: http://dx.doi.org/10.1016/j.nonrwa.2008.09.019.
    Rational solutions of the classical Boussinesq system are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation, which arises as a scaling reduction of the classical Boussinesq system. Generalized rational solutions of the classical Boussinesq system, which involve an infinite number of arbitrary constants, are also derived. The generalized rational solutions are analogues of such solutions for the Korteweg–de Vries, Boussinesq and nonlinear Schrödinger equations.
  • Clarkson, P. and Filipuk, G. (2008). The symmetric fourth Painleve hierarchy and associated special polynomials. Studies in Applied Mathematics [Online] 121:157-188. Available at: http://dx.doi.org/10.1111/j.1467-9590.2008.00410.x.
    In this paper two families of rational solutions and associated special polynomials for the equations in the symmetric fourth Painlevé hierarchy are studied. The structure of the roots of these polynomials is shown to be highly regular in the complex plane. Further representations are given of the associated special polynomials in terms of Schur functions. The properties of these polynomials are compared and contrasted with the special polynomials associated with rational solutions of the fourth Painlevé equation.
  • Clarkson, P. (2008). Rational Solutions Of The Boussinesq Equation. Analysis and Applications [Online] 6:349-369. Available at: http://dx.doi.org/10.1142/S0219530508001250.
    Rational solutions of the Boussinesq equation are expressed in terms of special polynomials associated with rational solutions of the second and fourth Painlevé equations, which arise as symmetry reductions of the Boussinesq equation. Further generalized rational solutions of the Boussinesq equation, which involve an infinite number of arbitrary constants, are derived. The generalized rational solutions are analogs of such solutions for the Korteweg–de Vries and nonlinear Schrödinger equations.
  • Harris, S. and Clarkson, P. (2006). Painlevé Analysis and Similarity Reductions for the Magma Equation. Symmetry, Integrability and Geometry: Methods and Applications [Online] 2. Available at: http://dx.doi.org/doi:10.3842/SIGMA.2006.068.
    In this paper, we examine a generalized magma equation for rational values of two parameters, m and n. Firstly, the similarity reductions are found using the Lie group method of infinitesimal transformations. The Painlevé ODE test is then applied to the travelling wave reduction, and the pairs of m and n which pass the test are identified. These particular pairs are further subjected to the ODE test on their other symmetry reductions. Only two cases remain which pass the ODE test for all such symmetry reductions and these are completely integrable. The case when m = 0, n = ?1 is related to the Hirota-Satsuma equation and for m = ½, n = ?½, it is a real, generalized, pumped Maxwell-Bloch equation.
  • Sen, A., Hone, A. and Clarkson, P. (2006). On the Lax pairs of symmetric Painleve equations. Studies in Applied Mathematics [Online] 117:299-319. Available at: http://dx.doi.org/10.1111/j.1467-9590.2006.00356.x.
    The symmetric forms of the Painlevé equations are a sequence of nonlinear dynamical systems in N+ 1 variables that admit the action of an extended affine Weyl group of type, as shown by Noumi and Yamada. They are equivalent to the periodic dressing chains studied by Veselov and Shabat, and by Adler. In this paper, a direct derivation of the symmetries of a corresponding sequence of (N+ 1) × (N+ 1) matrix linear systems (Lax pairs) is given. The action of the generators of the extended affine Weyl group of type on the associated Lax pairs is realized through a set of transformations of the eigenfunctions, and this extends to an action of the whole group.
  • Bila, N., Mansfield, E. and Clarkson, P. (2006). Symmetry group analysis of the shallow water and semi-geostrophic equations. Quarterly Journal of Mechanics and Applied Mathematics [Online] 59:95-123. Available at: http://dx.doi.org/10.1093/qjmam/hbi033.
    The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semi-geostrophic equations in such a way that a well-defined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an infinite linear cascade'.
  • Clarkson, P. et al. (2006). One hundred years of PVI, the Fuchs–Painlevé equation - Preface. Journal of Physics A: Mathematical and General [Online] 39. Available at: http://dx.doi.org/10.1088/0305-4470/39/39/E01.
  • Clarkson, P. (2006). Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations. Computational Methods and Function Theory 6:329-401.
    Rational solutions of the second, third and fourth Painlev´e equations can be expressed in terms of
    special polynomials defined through second order, bilinear differential-difference equations which are equivalent
    to the Toda equation. In this paper the structure of the roots of these special polynomials, as well as the special
    polynomials associated with algebraic solutions of the third and fifth Painlev´e equations and equations in the
    PII hierarchy, are studied. It is shown that the roots of these polynomials have an intriguing, highly symmetric
    and regular structure in the complex plane. Further, using the Hamiltonian theory for the Painlev´e equations,
    other properties of these special polynomials are studied. Soliton equations, which are solvable by the inverse
    scattering method, are known to have symmetry reductions which reduce them to Painlev´e equations. Using this
    relationship, rational solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations and rational
    and rational-oscillatory solutions of the non-linear Schr¨odinger equation are expressed in terms of these special polynomials.
  • Clarkson, P. (2006). Special polynomials associated with rational solutions of the defocusing nonlinear Schrodinger equation and the fourth Painleve equation. European Journal of Applied Mathematics [Online] 17:293-322. Available at: http://dx.doi.org/10.1017/S0956792506006565.
    Rational solutions and rational-oscillatory solutions of the defocusing nonlinear Schrödinger equation are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation. The roots of these special polynomials have a regular, symmetric structure in the complex plane. The rational solutions verify results of Nakamura and Hirota [J. Phys. Soc. Japan, 54 (1985) 491–499] whilst the rational-oscillatory solutions appear to be new solutions of the defocusing nonlinear Schrödinger equation.
  • Sen, A., Hone, A. and Clarkson, P. (2005). Darboux transformations and the symmetric fourth Painlevé equation. Journal of Physics A: Mathematical and General [Online] 38:9751-9764. Available at: http://dx.doi.org/10.1088/0305-4470/38/45/003.
    This paper is concerned with the group symmetries of the fourth Painleve equation P-IV, a second-order nonlinear ordinary differential equation. It is well known that the parameter space of P-IV admits the action of the extended affine Weyl group A(2)((1)). As shown by Noumi and Yamada, the action of A(2)((1)) as Backlund transformations of P-IV provides a derivation of its symmetric form SP4. The dynamical System SP4 is also equivalent to the isomonodromic deformation of an associated three-by-three matrix linear system (Lax pair). The action of the generators of A(2)((1)) on this Lax pair is derived using the Darboux transformation for an associated third-order operator
  • Clarkson, P. (2005). Special polynomials associated with rational solutions of the fifth Painlevé equation. Journal of Computational and Applied Mathematics [Online] 178:111-129. Available at: http://dx.doi.org/10.1016/j.cam.2004.04.015.
    In this paper special polynomials associated with rational and algebraic solutions of the fifth Painlevéequation (PV) are studied. These special polynomials defined by second-order, bilinear differential-difference equations which are equivalent to Toda equations. The structure of the zeroes of these special polynomials, which involve a parameter, is investigated and it is shown that these have an intriguing, symmetric and regular structure. For large negative values of the parameter the zeroes have an approximate triangular structure. As the parameter increases the zeroes coalesce for certain values and eventually for large positive values of the parameter the zeroes also have an approximate triangular structure, though with the orientation reversed. In fact, the interaction of the zeroes is “solitonic” in nature since the same pattern reappears, with its orientation reversed.
  • Clarkson, P. (2003). Remarks on the Yablonskii–Vorob'ev polynomials. Physics Letters A [Online] 319:137-144. Available at: http://dx.doi.org/10.1016/j.physleta.2003.10.016.
    It is well known that rational solutions of the second Painlevé equation (PII) are expressed in terms of logarithmic derivatives of the Yablonskii–Vorob'ev polynomials Qn(z) which are defined through a second order, bilinear differential-difference equation which is equivalent to the Toda equation. In this Letter, using the Hamiltonian theory for PII, it is shown that Qn(z) also satisfies a fourth order, bilinear ordinary differential equation and a fifth order, quad-linear difference equation. Further, rational solutions of some ordinary differential equations which are solvable in terms of solutions of PII are also expressed in terms of the Yablonskii–Vorob'ev polynomials.
  • Clarkson, P. (2003). The fourth Painlevé equation and associated special polynomials. Journal of Mathematical Physics [Online] 44:5350-5374. Available at: http://dx.doi.org/10.1063/1.1603958.
    In this article rational solutions and associated polynomials for the fourth Painlevé equation are studied. These rational solutions of the fourth Painlevé equation are expressible as the logarithmic derivative of special polynomials, the Okamoto polynomials. The structure of the roots of these Okamoto polynomials is studied and it is shown that these have a highly regular structure. The properties of the Okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials. Further representations are given of the associated rational solutions in the form of determinants through Schur functions
  • Clarkson, P., Hone, A. and Joshi, N. (2003). Hierarchies of Difference Equations and Bäcklund Transformations. Journal of Nonlinear Mathematical Physics [Online] 10:13-26. Available at: http://dx.doi.org/10.2991/jnmp.2003.10.s2.2.
    In this paper we present a method for deriving infinite sequences of difference equations containing well known discrete Painlevé equations by using the Bäcklund transformtions for the equations in the second Painlevé equation hierarchy.
  • Clarkson, P. (2003). The third Painlevé equation and associated special polynomials. Journal of Physics A: Mathematical and General [Online] 36:9507-9532. Available at: http://dx.doi.org/10.1088/0305-4470/36/36/306.
    In this paper we are concerned with rational solutions, algebraic solutions and associated special polynomials with these solutions for the third Painlevé equation (PIII). These rational and algebraic solutions of PIII are expressible in terms of special polynomials defined by second-order, bilinear differential-difference equations which are equivalent to Toda equations. The structure of the roots of these special polynomials is studied and it is shown that these have an intriguing, highly symmetric and regular structure. Using the Hamiltonian theory for PIII, it is shown that these special polynomials satisfy pure difference equations, fourth-order, bilinear differential equations as well as differential-difference equations. Further, representations of the associated rational solutions in the form of determinants through Schur functions are given.
  • Clarkson, P. (2003). Painlevé equations—nonlinear special functions. Journal of Computational and Applied Mathematics [Online] 153:127-140. Available at: http://dx.doi.org/10.1016/S0377-0427(02)00589-7.
    The six Painlevé equations (PI–PVI) were first discovered about a hundred years ago by Painlevéand his colleagues in an investigation of nonlinear second-order ordinary differential equations. Recently, there has been considerable interest in the Painlevé equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently, the Painlevé equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations, despite being nonlinear equations. Although first discovered from strictly mathematical considerations, the Painlevé equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics.

    The Painlevé equations may be thought of a nonlinear analogues of the classical special functions. They possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further the Painlevé equations admit symmetries under affine Weyl groups which are related to the associated Bäcklund transformations.

    In this paper, I discuss some of the remarkable properties which the Painlevé equations possess including connection formulae, Bäcklund transformations associated discrete equations, and hierarchies of exact solutions. In particular, the second Painlevé equation PII is used to illustrate these properties and some of the applications of PII are also discussed.
  • Clarkson, P. and Mansfield, E. (2003). The second Painleve equation, its hierarchy and associated special polynomials. Nonlinearity [Online] 16:R1-R26. Available at: http://dx.doi.org/10.1088/0951-7715/16/3/201.
    In this paper we are concerned with hierarchies of rational solutions and associated polynomials for the second Painleve equation (P-II) and the equations in the P-II hierarchy which is derived from the modified Korteweg-de Vries hierarchy. These rational solutions of P-II are expressible as the logarithmic derivative of special polynomials, the Yablonskii-Vorob'ev polynomials. The structure of the roots of these Yablonskii-Vorob'ev polynomials is studied and it is shown that these have a highly regular triangular structure. Further, the properties of the Yablonskii-Vorob'ev polynomials are compared and contrasted with those of classical orthogonal polynomials. We derive the special polynomials for the second and third equations of the P-II hierarchy and give a representation of the associated rational solutions in the form of determinants through Schur functions. Additionally the analogous special polynomials associated with rational solutions and representation in the form of determinants are conjectured for higher equations in the P-II hierarchy. The roots of these special polynomials associated with rational solutions for the equations of the P-II hierarchy also have a highly regular structure.
  • Hu, X. and Clarkson, P. (2002). Rational Solutions of an Extended Lotka-Volterra Equation. Journal of Nonlinear Mathematical Physics [Online] 9:75-93. Available at: http://dx.doi.org/10.2991/jnmp.2002.9.s1.7.
    A series of rational solutions are presented for an extended Lotka-Volterra equation. These rational solutions are obtained by using Hirota's bilinear formalism and Backlund transformation. The crucial step is the use of nonlinear superposition formula.

Book section

  • Clarkson, P. (2008). The fourth Painleve equation. in: Guo, L. and Sit, W. Y. eds. Differential Algebra and Related Topics. Singapore: World Scientific. Available at: http://www.worldscibooks.com/mathematics/6969.html.
    The six Painleve equations (PI–PVI) were first discovered about a hundred years ago by Painleve and his colleagues in an investigation of nonlinear second-order ordinary differential equations. During the past 30 years there has been considerable interest in the Painleve equations primarily due to the fact that they arise as symmetry reductions of the soliton equations which are solvable by inverse scattering. Although first discovered from pure mathematical considerations, the Painleve equations have arisen in a variety of important physical applications.

    The Painleve equations may be thought of as nonlinear analogues of the classical special functions. They have a Hamiltonian structure and associated isomonodromy problems, which express the Painleve equations as the compatibility condition of two linear systems. The Painleve equations also admit symmetries under affine Weyl groups which are related to the associated B¨acklund transformations. These can be used to generate hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions, for special values of the parameters. Further solutions of the Painleve equations have some interesting asymptotics which are use in applications. In this paper I discuss some of the remarkable properties which the Painleve equations possess using the fourth Painleve equation (PIV) as an illustrative example.
  • Clarkson, P., Joshi, N. and Mazzocco, M. (2006). The Lax pair for the mKdV hierarchy. in: Delabaere, E. and Loday-Richaud, M. eds. Théories Asymptotiques et Equations de Painlevé. Paris, France: Sociètè Mathèematique de France, pp. 53-64.
    Rational solutions of the second, third and fourth Painlevé equations (-) can be expressed in terms of logarithmic derivatives of special polynomials that are defined through coupled second order, bilinear differential-difference equations which are equivalent to the Toda equation.

    In this paper the structure of the roots of these special polynomials, and the special polynomials associated with algebraic solutions of the third and fifth Painlevé equations, is studied and it is shown that these have an intriguing, highly symmetric and regular structure. Further, using the Hamiltonian theory for -, it is shown that all these special polynomials, which are defined by differential-difference equations, also satisfy fourth order, bilinear ordinary differential equations.
  • Clarkson, P. (2006). Painleve equations - nonlinear special functions. in: Marcellan, F. and van Asschel, W. eds. Orthogonal Polynomials and Special Functions: Computation and Application. Berlin/ Heidelberg: Springer-Verlag, pp. 331-411. Available at: http://dx.doi.org/10.1007/978-3-540-36716-1_7.
  • Clarkson, P. (2006). Special polynomials associated with rational and algebraic solutions of the Painleve equations. in: Delabaere, E. and Loday-Richaud, M. eds. Theories Asymptotiques et Equations de Painleve. Paris, France: Societe Mathematique de France, pp. 21-52.
    Rational solutions of the second, third and fourth Painlevé equations (-) can be expressed in terms of logarithmic derivatives of special polynomials that are defined through coupled second order, bilinear differential-difference equations which are equivalent to the Toda equation.

    In this paper the structure of the roots of these special polynomials, and the special polynomials associated with algebraic solutions of the third and fifth Painlevé equations, is studied and it is shown that these have an intriguing, highly symmetric and regular structure. Further, using the Hamiltonian theory for -, it is shown that all these special polynomials, which are defined by differential-difference equations, also satisfy fourth order, bilinear ordinary differential equations.
  • Clarkson, P. (2005). Painlevé Equations and Associated Polynomials. in: Ismail, M. E. H. and Koelink, E. eds. Theory and Applications of Special Functions: A Volume Dedicated to Mizan Rahman. Berlin: Springer, pp. 123-163. Available at: http://dx.doi.org/10.1007/0-387-24233-3_7.
    In this paper we are concerned with rational solutions and associated polynomials for the second, third and fourth Painlevé equations. These rational solutions are expressible as in terms of special polynomials. The structure of the roots of these polynomials is studied and it is shown that these have a highly regular structure.
  • Clarkson, P. (2003). On rational solutions of the fourth Painleve equation and its Hamiltonian. in: Winternitz, P. et al. eds. Group Theory and Numerical Analysis. United States: American Mathematical Society, pp. 103-118.
  • Clarkson, P., Mansfield, E. and Webster, H. (2002). On Discrete Painleve Equations as Backlund Transformations. in: Coley, A. et al. eds. Backlund and Darboux Transformations: The Geometry of Solitons. United States: American Mathematical Society, pp. 129-139.
  • Clarkson, P. and Mansfield, E. (2002). Open problems in symmetry analysis. in: Leslie, J. ed. The Geometrical Study of Differential Equations. United Kingdom: American Mathematical Society, pp. 195-205.

Conference or workshop item

  • Clarkson, P. (2008). Asymptotics of the second Painleve equation. in: Dominici, D. and Maier, R. eds. Providence, RI, USA: American Mathematical Society, pp. 69-83.

Edited journal

  • Clarkson, P. and Daalhuis, A.B.O. eds. (2018). Special Issue: Orthogonal Polynomials, Special Functions and Applications. Studies in Applied Mathematics [Online] 141. Available at: https://onlinelibrary.wiley.com/toc/14679590/2018/141/4.

Monograph

  • Rogers, C. and Clarkson, P. (2017). Ermakov-Painlevé II Symmetry Reduction of a Korteweg Capillarity System. arXiv.org. Available at: https://arxiv.org/abs/1701.03238.
    A class of nonlinear Schr\"{o}dinger equations involving a triad of power law terms together with a de Broglie-Bohm potential is shown to admit symmetry reduction to a hybrid Ermakov-Painlev\'{e} II equation which is linked, in turn, to the integrable Painlev\'{e} XXXIV equation. A nonlinear Schr\"{o}dinger encapsulation of a Korteweg-type capillary system is thereby used in the isolation of such a Ermakov-Painlev\'{e} II reduction valid for a multi-parameter class of free energy functions. Iterated application of a B\"{a}cklund transformation then allows the construction of novel classes of exact solutions of the nonlinear capillarity system in terms of Yablonskii-Vorob'ev polynomials or classical Airy functions. A Painlev\'{e} XXXIV equation is derived for the density in the capillarity system and seen to correspond to the symmetry reduction of its Bernoulli integral of motion.
  • Clarkson, P., Law, C. and Lin, C. (2016). An algebraic proof for the Umemura polynomials for the third Painlevé equation. arxiv.org. Available at: https://arxiv.org/abs/1609.00495.
    We are concerned with the Umemura polynomials associated with the third Painlev\'e equation. We extend Taneda's method, which was developed for the Yablonskii-Vorob'ev polynomials associated with the second Painlev\'e equation, to give an algebraic proof that the rational functions generated by the nonlinear recurrence relation satisfied by Umemura polynomials are indeed polynomials.
  • Ankiewicz, A. et al. (2016). Conservation laws and integral relations for the Boussinesq equation. arXiv.org. Available at: https://arxiv.org/abs/1611.09505.
    We are concerned with conservation laws and integral relations associated with rational solutions of the Boussinesq equation, a soliton equation solvable by inverse scattering which was first introduced by Boussinesq in 1871. The rational solutions are logarithmic derivatives of a polynomial, are algebraically decaying and have a similar appearance to rogue-wave solutions of the focusing nonlinear Schr\"{o}dinger equation. For these rational solutions the constants of motion associated with the conserved quantities are zero and they have some interesting integral relations which depend on the total degree of the associated polynomial.
  • Clarkson, P. and Jordaan, K. (2016). Properties of Generalized Freud Polynomials. arxiv.org. Available at: https://arxiv.org/abs/1606.06026.
    We consider the semi-classical generalized Freud weight function
    w?(x;t)=|x|2?+1exp(?x4+tx2),x??,
    with ?>?1 and t?? parameters. We analyze the asymptotic behavior of the sequences of monic polynomials that are orthogonal with respect to w?(x;t), as well as the asymptotic behavior of the recurrence coefficient, when the degree, or alternatively, the parameter t, tend to infinity. We also investigate existence and uniqueness of positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients and prove properties of the zeros of the generalized Freud polynomials.
  • Clarkson, P. and Dowie, E. (2016). Rational solutions of the Boussinesq equation and applications to rogue waves. arxiv.org. Available at: https://arxiv.org/abs/1609.00503.
    We study rational solutions of the Boussinesq equation, which is a soliton equation solvable by the inverse scattering method. These rational solutions, which are algebraically decaying and depend on two arbitrary parameters, are expressed in terms of special polynomials that are derived through a bilinear equation, have a similar appearance to rogue-wave solutions of the focusing nonlinear Schr\"{o}dinger (NLS) equation and have an interesting structure. Further rational solutions of the Kadomtsev-Petviashvili I (KPI) equation are derived in two ways, from rational solutions of the NLS equation and from rational solutions of the Boussinesq equation. It is shown that the two families of rational solutions of the KPI equation are fundamentally different.

Forthcoming

  • Clarkson, P. et al. eds. (2018). Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) - Special Issue on Orthogonal Polynomials, Special Functions and Applications. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) - Special Issue on Orthogonal Polynomials, Special Functions and Applications [Online]. Available at: https://www.emis.de/journals/SIGMA/OPSFA2017.html.
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