# Rational Solutions of Soliton equations and Applications to Vortex Dynamics

*by Professor Peter Clarkson*

*Institution:*University of Kent, UK

*Date: Tue 8th September 2009*

*Time: 1:00 PM*

*Location: Room 213, Richard Berry Building, The University of Melbourne*

*Abstract*: In this talk I shall discuss special polynomials associated with rational solutions for the Painleve equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations.

The Painleve equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions. Rational solutions of the Painleve equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the second Painleve equation (PII) these polynomials are known as the Yablonskii{Vorob'ev polynomials, first derived in the 1960's by Yablonskii and Vorob'ev. The locations of the roots of these polynomials is shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials associated with rational solutions of the fourth Painleve equation (PIV), which are known as the generalized Hermite polynomials and generalized Okamoto polynomials, are described and it is shown that their roots also have a highly regular structure. The Yablonskii Vorob'ev polynomials arise in string theory and the generalized Hermite polynomials in the theories of random matrices and orthogonal polynomials.

It is well known that soliton equations have symmetry reductions which reduce them to the Painleve equations, e.g. scaling reductions of the Korteweg-de Vries equation is expressible in terms of PII and scaling reductions of the Boussinesq and nonlinear Schrodinger equations are expressible in terms of PIV. Hence rational solutions of these soliton equations can be expressed in terms of the Yablonskii and Vorob'ev, generalized Hermite and generalized Okamoto polynomials. Further general rational solutions of equations for the Korteweg-de Vries, Boussinesq equations and nonlinear Schrodinger equations, which involve arbitrary parameters, will also be described.

Finally I shall discuss applications of these special polynomials associated with rational solutions for the Painleve and soliton equations to point vortex dynamics.

*For More Information:* contact: Nicholas Witte. email: N.Witte@ms.unimelb.edu.au