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SUMMARY:Vortices\, rogue waves and polynomials - Peter Clarkson (Universit
 y of Kent)
DTSTART:20110310T150000Z
DTEND:20110310T160000Z
UID:TALK29918@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:In this talk I shall discuss special polynomials associated wi
 th rational solutions of the Painlevé equations and of the soliton equati
 ons which are solvable by the inverse scattering method\, including the Ko
 rteweg-de Vries\, Boussinesq and nonlinear Schrödinger equations. Further
  I shall illustrate applications of these polynomials to vortex dynamics a
 nd rogue waves.\n\nThe Painlevé equations are six nonlinear ordinary diff
 erential equations that have been the subject of much interest in the past
  thirty years\, and have arisen in a variety of physical applications. Fur
 ther the Painlevé equations may be thought of as nonlinear special functi
 ons. Rational solutions of the Painlevé equations are expressible in term
 s of the logarithmic derivative of certain special polynomials. For the fo
 urth Painlevé equation these polynomials are known as the _generalized He
 rmite polynomials_ and _generalized Okamoto polynomials_. The locations of
  the roots of these polynomials have a highly symmetric (and intriguing) s
 tructure in the complex plane. \n\nIt 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 equati
 ons are expressible in terms of the fourth Painlevé equation. Hence ratio
 nal solutions of these equations can be expressed in terms of the generali
 zed Hermite and generalized Okamoto polynomials. \n\nI will also discuss t
 he 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. Stat
 ionary vortex configurations with vortices of the same strength and positi
 ve or negative configurations are located at the roots of the _Adler-Moser
  polynomials_\, which are associated with rational solutions of the Kortwe
 g-de Vries equation. \n\nFurther\, I shall also describe some additional r
 ational solutions of the Boussinesq equation and and rational-oscillatory 
 solutions of the focusing nonlinear Schrödinger equation which have appli
 cations to rogue waves.\n
LOCATION:MR14\, CMS
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