University of Kent

This session will discuss how Douglas Hartree and Arthur Porter used Meccano — a child’s toy and an engineer’s tool — to build an analogue computer, the Hartree Differential Analyser in 1934. It will explore the wider historical and social context in which this model computer was rooted, before providing an opportunity to engage with the experiential aspects of the 'Kent Machine,' a historically reproduced version of Hartree and Porter's original model, which is also made from Meccano.

The 'Kent Machine' sits at a unique intersection of historical research and educational engagement, providing an alternative way of teaching STEM subjects, via a historic hands-on method. The session builds on the work and ideas expressed in Otto Sibum's reconstruction of James Joule's 'Paddle Wheel' apparatus, inviting attendees to physically re-enact the mathematical processes of mechanical integration to see how this type of analogue computer functioned in reality. The session will provide an alternative context of the history of computing by exploring the tacit knowledge that is required to reproduce and demonstrate the machine, and how it sits at the intersection between amateur and professional science.

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.