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University of Cambridge > Talks.cam > Applied and Computational Analysis > Vortices, rogue waves and polynomials

## Vortices, rogue waves and polynomialsAdd to your list(s) Download to your calendar using vCal - Peter Clarkson (University of Kent)
- Thursday 10 March 2011, 15:00-16:00
- MR14, CMS.
If you have a question about this talk, please contact ai10. In this talk I shall discuss special polynomials associated with rational solutions of the Painlevé equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrödinger equations. Further I shall illustrate applications of these polynomials to vortex dynamics and rogue waves. The Painlevé equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, and have arisen in a variety of physical applications. Further the Painlevé equations may be thought of as nonlinear special functions. Rational solutions of the Painlevé equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the fourth Painlevé equation these polynomials are known as the It is well known that soliton equations have symmetry reductions which reduce them to the Painlevé equations, e.g. scaling reductions of the Boussinesq and nonlinear Schrödinger equations are expressible in terms of the fourth Painlevé equation. Hence rational solutions of these equations can be expressed in terms of the generalized Hermite and generalized Okamoto polynomials. I will also discuss the relationship between vortex dynamics and properties of polynomials with roots at the vortex positions. Classical polynomials such as the Hermite and Laguerre polynomials have roots which describe vortex equilibria. Stationary vortex configurations with vortices of the same strength and positive or negative configurations are located at the roots of the Further, I shall also describe some additional rational solutions of the Boussinesq equation and and rational-oscillatory solutions of the focusing nonlinear Schrödinger equation which have applications to rogue waves. This talk is part of the Applied and Computational Analysis series. ## This talk is included in these lists:- All CMS events
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