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CATEGORIES:Applied and Computational Analysis
SUMMARY:Vortices\, rogue waves and polynomials - Peter Cla
rkson (University of Kent)
DTSTART;TZID=Europe/London:20110310T150000
DTEND;TZID=Europe/London:20110310T160000
UID:TALK29918AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/29918
DESCRIPTION:In this talk I shall discuss special polynomials a
ssociated with rational solutions of the Painlevé
equations and of the soliton equations which are s
olvable by the inverse scattering method\, includi
ng the Korteweg-de Vries\, Boussinesq and nonlinea
r Schrödinger equations. Further I shall illustrat
e applications of these polynomials to vortex dyna
mics and rogue waves.\n\nThe Painlevé equations ar
e six nonlinear ordinary differential equations th
at have been the subject of much interest in the p
ast thirty years\, and have arisen in a variety of
physical applications. Further the Painlevé equat
ions may be thought of as nonlinear special functi
ons. Rational solutions of the Painlevé equations
are expressible in terms of the logarithmic deriva
tive of certain special polynomials. For the fourt
h Painlevé equation these polynomials are known as
the _generalized Hermite polynomials_ and _genera
lized Okamoto polynomials_. The locations of the r
oots of these polynomials have a highly symmetric
(and intriguing) structure in the complex plane. \
n\nIt is well known that soliton equations have sy
mmetry reductions which reduce them to the Painlev
é equations\, e.g. scaling reductions of the Bouss
inesq and nonlinear Schrödinger equations are expr
essible 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. \n\nI will also
discuss the relationship between vortex dynamics a
nd properties of polynomials with roots at the vor
tex positions. Classical polynomials such as the H
ermite and Laguerre polynomials have roots which d
escribe vortex equilibria. Stationary vortex confi
gurations with vortices of the same strength and p
ositive or negative configurations are located at
the roots of the _Adler-Moser polynomials_\, which
are associated with rational solutions of the Kor
tweg-de Vries equation. \n\nFurther\, I shall also
describe some additional rational solutions of th
e Boussinesq equation and and rational-oscillatory
solutions of the focusing nonlinear Schrödinger e
quation which have applications to rogue waves.\n
LOCATION:MR14\, CMS
CONTACT:
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