Special Applied Maths Seminar: Clarkson -- Rational Solutions of Soliton Equations and Applications to Vortex Dynamics

Special Applied Maths Seminar (note the unusual time)

Peter Clarkson, University of Kent 

will talk about 

Rational Solutions of Soliton Equations and Applications to Vortex Dynamics 

on Thursday, 27 August, 2pm, Eastern Avenue Lecture Theatre 

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

The Painlev´e equations are six nonlinear ordinary differential equations that
have been the sub ject 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 Painlev´e equations are expressible in terms of
the logarithmic derivative of certain special polynomials.  For the second Painlev´e
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 Painlev´e 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 Painlev´e equations, e.g.  scaling reductions of the Korteweg-de Vries equation is
express- ible in terms of PII and scaling reductions of the Boussinesq and nonlinear
Schr¨ odinger equa- tions 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
Schr¨odinger equations, which involve arbitrary parameters, will also be described.  

I shall also discuss applications of these special polynomials associated with rational
so- lutions for the Painlev´e and soliton equations to point vortex dynamics.  Further
multivortex solutions of the complex Sine-Gordon equation on the plane will be expressed
in terms of special polynomials associated with rational solutions of the fifth
Painlev´e equation, which are expressed as double wronskians of associated Laguerre
polynomials.