Type: Seminar

Distribution: World

Expiry: 28 Aug 2009

Auth: dullin@p714.pc (assumed)

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.