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 and discrete Painlevé equations.
• Symmetry reductions and exact solutions of nonlinear partial differential equations, in particular using nonclassical and generalized techniques.
• Symbolic manipulation programmes for symmetry analysis.
• Reductions of the self-dual Yang-Mills equations.

Current Research Project

• I am participant in the Digital Library of Mathematical Functions project, 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. My role in the project is writing the chapter on Painlevé Transcendents.

SIAM Activity Group on Orthogonal Polynomials and Special Functions

• Currently I am the vice-chair of the SIAM Activity Group on Orthogonal Polynomials and Special Functions
  
• The activity group promotes basic research in orthogonal polynomials and special functions; furthers the application of this subject in other parts of mathematics, and in science and industry; and encourages and supports the exchange of information, ideas, and techniques between workers in this field and other mathematicians and scientists.
  
• The activity group publishes the electronic newsletter OP-SF NET, and maintains a web site which includes a conference calendar.
  

Forthcoming Conferences

• Joint Mathematics Meeting, San Francesco, CA, USA (13th-16th January 2010)
  
• South African Symposium on Numerical and Applied Mathematics 2010, Stellenbosch, South Africa (15th-17th March 2010)
  
• Maths2010, Joint BMC/BAMC Meeting, Edinburgh (6th-9th April 2010)
  
• Numerical Solutions of Painlevé Equations, Edinburgh (10th-14th May 2010)
  
• Group Analysis of Differential Equations and Integrable Systems, Protaras, Cyprus (6th-10th June 2010)
  
• Nonlinear Waves-Theory and Applications, Beijing, China (26th-29th June 2010)
  
• Symmetry, Separation, Super-integrability and Special Functions, Minneapolis, MN, USA (17th-19th September 2010)
  
• Foundations of Computational Mathematics 2011, Budapest, Hungary (4th-14th July 2011)
  
• 7th International Congress on Industrial and Applied Mathematics (ICIAM 2011), Vancouver, Canada (18th-22nd July 2011)
  

Forthcoming Research Programmes

• Random Matrix Theory, Interacting Particle Systems and Integrable Systems, Mathematical Sciences Research Institute, Berkley, USA (16th August-17th December 2010)
  

Editorial Board Membership

• EqWorld: The World of Mathematical Equations
  
• European Journal of Applied Mathematics
  
• International Journal of Mathematical Analysis
  
• Journal of Nonlinear Mathematical Physics
  
• Proceedings of the American Mathematical Society, Coordinating Editor for Applied Mathematics, Probability and Statistics
  
• Quarterly Journal of Mechanics and Applied Mathematics
  
• Studies in Applied Mathematics
• Symmetry, Integrability and Geometry: Methods and Applications
  

Publications

Papers in Conference Proceedings

1. P A Clarkson and J B McLeod, "A connection formula for the second Painlevé transcendent", in Ordinary and Partial Differential Equations [Editors W N Everitt and B D Sleeman], Lecture Notes in Mathematics vol. 964, Springer-Verlag, Berlin (1983) pp 135-142.
2. P A Clarkson, "The Painlevé conjecture, the Painlevé property for partial differential equations and complete integrability", in Solitons and Coherent Structures [Editors D K Campbell, A C Newell, R J Schrieffer and H Segur], Physica D18 (1986) pp 209-210.
3. P A Clarkson, "New similarity reductions of Boussinesq-type equations", in Partially Integrable Nonlinear Evolution Equations and their Physical Applications [Editors R Conte and N Boccara], NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, vol. 310, Kluwer, Dordrecht (1990) pp 575-576.
4. P A Clarkson, "Examples of nonclassical similarity reductions", in Nonlinear Evolution Equations and Dynamical Systems [Editors S Carillo and O Ragnisco], Research Reports in Physics (Nonlinear Dynamics), Springer-Verlag, Berlin-Heidelberg (1990) pp 42-45.
5. P A Clarkson, "Nonclassical symmetry reductions and exact solutions for physically significant nonlinear evolution equations", in Nonlinear and Chaotic Phenomena [Editors W Rozmus and J A Tuszynski], World Scientific, Singapore (1991) pp 72-79.
6. P A Clarkson, "Nonclassical symmetry reductions of a generalized nonlinear Schrödinger equation", in Solitons and Chaos [Editors I Antoniou and F Lambert], Research Reports in Physics (Nonlinear Dynamics), Springer-Verlag, Berlin (1991) pp 246-254.
7. S Chakravarty, M J Ablowitz and P A Clarkson, "One dimensional reductions of self-dual Yang-Mills fields and classical systems", in Recent Advances in General Relativity: Essays in Honor of Ted Newman [Editors A I Janis and J R Porter], Birkhäuser Boston, Basel (1992) pp 60-71.
8. P A Clarkson and J B McLeod, "Integral equations and connection formulae for the Painlevé equations", in Painlevé Transcendents, their Asymptotics and Physical Applications [Editors P Winternitz and D Levi], NATO Advanced Study Institute Series B: Physics, vol. 278, Plenum, New York (1992) pp 1-31.
9. P A Clarkson and E L Mansfield, "Nonclassical symmetry reductions and exact solutions of nonlinear reaction-diffusion equations", in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations [Editor P A Clarkson], NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, vol. 413, Kluwer, Dordrecht (1993) pp 375-389.
10. A C Hicks, P A Clarkson and A P Bassom, "A study of the fourth Painlevé equation", in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations [Editor P A Clarkson], NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, vol. 413, Kluwer, Dordrecht (1993) pp 315-330.
11. S Hood and P A Clarkson, "Symmetry reductions of the Davey-Stewartson system", in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations [Editor P A Clarkson], NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, vol. 413, Kluwer, Dordrecht (1993) pp 395-404.
12. D K Ludlow and P A Clarkson, "Symmetry reductions and exact solutions for a generalized Boussinesq equation", in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations [Editor P A Clarkson], NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, vol. 413, Kluwer, Dordrecht (1993) pp 415-430.
13. A E Milne and P A Clarkson, "Rational solutions and Bäcklund transformations for the third Painlevé equation", in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations [Editor P A Clarkson], NATO Advanced Study Institute Series C: Mathematical and Physical Sciences, vol. 413, Kluwer, Dordrecht (1993) pp 341-352.
14. P A Clarkson and E L Mansfield, "Symmetries of the nonlinear heat equation", in Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics [Editors N H Ibragimov, M Torrisi and A Valenti], Kluwer, Dordrecht (1993) pp 155-171.
15. P A Clarkson and E L Mansfield, "Symmetry reductions and exact solutions of a shallow water wave equation", in Fourteenth IMACS World Congress on Computational and Applied Mathematics [Editor W F Ames], 1994, vol. 1, IMACS (1994) pp 97-101.
16. E L Mansfield and P A Clarkson, "Applications of the differential algebra package diffgrob2 to reductions of PDE", in Fourteenth IMACS World Congress on Computational and Applied Mathematics [Editor W F Ames], vol. 1, IMACS (1994) pp 336-339.
17. P A Clarkson and E L Mansfield, "Symmetry reductions and exact solutions of shallow water wave equations", in Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology [Editors D B Duncan and J C Eilbeck], published on WWW (1995).
18. T J Priestley and P A Clarkson, "On a shallow water wave system", in Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology [Editors D B Duncan and J C Eilbeck], published on WWW (1995).
19. P A Clarkson, E L Mansfield and S B Townley, "Symmetries for nonlinear diffusion equations with forcing feedback", in Third European Control Conference (1995) pp 2911-2916.
20. P A Clarkson and A P Bassom, "Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation and their application to discrete equations", in Symmetries and Integrability of Difference Equations [Editors D Levi, L Vinet and P Winternitz], Centre de Recherches Mathématiques Proceedings and Lecture Notes Series, vol. 9, American Mathematical Society, Providence, RI (1996) pp 63-77.
21. X-B Hu and P A Clarkson, "Bäcklund transformation and new explicit solutions of the two-dimensional Toda equation", in Symmetries and Integrability of Difference Equations II [Editors P A Clarkson and F W Nijhoff], London Mathematical Society Lecture Note Series, vol. 255, Cambridge University Press, Cambridge (1999) pp 15-22.
22. P A Clarkson, "The Chazy equation", in Modern Group Analysis: Developments in theory computation and application [Editors N H Ibragimov, K R Naqvi and E Straume], MARS Publishers, Trondheim, Norway (1999) pp 79-89.
23. P A Clarkson and P Winternitz, "Symmetry reduction and exact solutions of nonlinear partial differential equations", in The Painlevé Property, One Century Later [Editor, R Conte], CRM Series in Mathematical Physics, Springer-Verlag, New York (1999) pp 591-660.
24. P G Estévez and P A Clarkson, "Discrete equations and the singular manifold method", in Symmetries and Integrability of Difference Equations III [Editors D Levi and P Winternitz], Centre de Recherches Mathématiques Proceedings and Lecture Notes Series, vol. 25, American Mathematical Society, Providence, RI (2000) pp 139-146.
25. P A Clarkson, E L Mansfield and H N Webster, "On discrete Painlevé equations as Bäcklund transformations", in Bäcklund and Darboux Transformations [Editors D Levi and P Winternitz], Centre de Recherches Mathématiques Proceedings and Lecture Notes Series, vol. 29, American Mathematical Society, Providence, RI (2001) pp 129-139.
26. P A Clarkson and E L Mansfield, "Open problems in symmetry analysis", in Geometrical Study of Differential Equations[Editors J A Leslie and T Robart], Contemporary Mathematics Series, vol. 285, American Mathematical Society, Providence, RI (2001) pp 195-205.
27. P A Clarkson and C M Cosgrove, "Symmetry, the Chazy equation and Chazy hierarchies", in Isomondromic Deformations and Applications in Physics [Editors J Harnad and A R Its], Centre de Recherches Mathématiques Proceedings and Lecture Notes Series, vol. 31, American Mathematical Society, Providence, RI (2002) pp 113-129.
28. P A Clarkson, "Painlevé equations and associated polynomials", in Theory and Applications of Special Functions. A volume dedicated to Mizan Rahman [Editors Mourad E H Ismail and Erik Koelink], Developments in Mathematics, vol. 13, Springer-Verlag, Berlin (2005) pp 123-163.
29. P A Clarkson, "Rational solutions of the fourth Painlevé equation and the nonlinear Schrödinger equation", in Group Analysis of Differential Equations, [Editors N H Ibragimov, C Sophocleous and P A Damianou], proceedings of the 10th International Conference in MOdern GRoup ANalysis (2005) pp 47-57.
30. P A Clarkson, "On rational solutions of the fourth Painlevé equation and its Hamiltonian", in Group Theory and Numerical Analysis [Editors P Winternitz, D Gomez-Ullate, A Iserles, D Levi, P J Olver, R Quispel and P Tempesta], Centre de Recherches Mathématiques Proceedings and Lecture Notes Series, vol. 39, American Mathematical Society, Providence, RI (2005) pp 103-118.
31. P A Clarkson, "Painlevé equations - nonlinear special functions", in Orthogonal Polynomials and Special Functions: Computation and Application [Editors F Márcellan and W van Assche], Lecture Notes in Mathematics, vol. 1883, Springer-Verlag, Berlin (2006) pp 331-411.
32. P A Clarkson, "Special polynomials associated with rational and algebraic solutions of the Painlevé equations", in Théories Asymptotiques et Equations de Painlevé [Editors E Delabaere and M Loday-Richaud], Séminaires et Congrès, vol. 14, Sociètè Mathèmatique de France, Paris (2007) pp 21-52.
33. P A Clarkson, N Joshi and M Mazzocco "The Lax pair for the mKdV hierarchy", in Théories Asymptotiques et Equations de Painlevé [Editors E Delabaere and M Loday-Richaud], Séminaires et Congrès, vol. 14, Sociètè Mathèmatique de France, Paris (2007) pp 53-64.
34. P A Clarkson "Asymptotics of the second Painlevé equation" in Special Functions and Orthogonal Polynomials [Editors Diego Dominici and Robert S Maier], Contemporary Mathematics Series, vol. 471, American Mathematical Society, Providence, RI (2008) pp 69-84.
35. P A Clarkson "The fourth Painlevé transcendent" in Differential Algebra and Related Topics [Editors Li Guo and William Y Sit], World Scientific, Singapore (2009) to appear.

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