Differential Equations and Applications Seminar
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Thu, 13/10/2011 16:00 |
Robert Mackay (University of Warwick) |
Differential Equations and Applications Seminar  |
DH 1st floor SR |
| Isostatic mounts are used in applications like telescopes and robotics to move and hold part of a structure in a desired pose relative to the rest, by driving some controls rather than driving the subsystem directly. To achieve this successfully requires an understanding of the coupled space of configurations and controls, and of the singularities of the mapping from the coupled space to the space of controls. It is crucial to avoid such singularities because generically they lead to large constraint forces and internal stresses which can cause distortion. In this paper we outline design principles for isostatic mount systems for dynamic structures, with particular emphasis on robots. | |||
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Thu, 20/10/2011 16:00 |
Alastair Rucklidge (University of Leeds) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Three-wave interactions form the basis of our understanding of many nonlinear pattern forming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, such as the Faraday wave experiment with multi-frequency forcing, consideration of three-wave interactions can explain the presence of the spatio-temporal chaos found in some experiments, enabling some previously unexplained results to be interpreted in a new light. The predictions are illustrated with numerical simulations of a model partial differential equation. | |||
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Thu, 27/10/2011 16:00 |
Peter Clarkson (University of Kent) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| 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 Schrodinger 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 generalized Hermite polynomials and generalized Okamoto polynomials. The locations of the roots of these polynomials have a highly symmetric (and intriguing) structure in the complex plane. 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 Adler-Moser polynomials, which are associated with rational solutions of the Kortweg-de Vries equation. Further, I shall also describe some additional rational solutions of the Boussinesq equation and rational-oscillatory solutions of the focusing nonlinear Schrödinger equation which have applications to rogue waves. | |||
