• Jun 13 2012

    Stellenbosch Node Seminar: Prof Bengt Fornberg (University of Colorado at Boulder, USA)

    ABSTRACT: The six Painlevé equations P_I to P_VI were introduced over a century ago, motivated by very theoretical considerations. Especially over the last several decades, these equations and their solutions - the Painlevé transcendents - have been found to play a central role in many areas of methematical physics, with the rapidly growing list now including statistical mechanics, combinatorics, plasma physics, nonlinear waves, quantum gravity, general relativity, string theory, Bose-Einstein condensates, Raman scattering, and nonlinear optics.

    Although the Painlevé equations have been the subject of extensive investigations for about a century, they have maintained a reputation of being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as 'numerical mine fields'. We note in this present work that, on the contrary, their unifying Painlevé property in fact provides excellent opportunities for very fast and accurate numerical solutions across the full complex plane. Our numerical method will be described for the P_I equation, with some solution illustrations given also for the P_IV equation.

    The numerical method development and calculations on P_I and P_II have been carried out in collaboration with Prof. André Weideman (University of Stellenbosch) and calculations on P_IV with Jonah Reeger (University of Colorado at Boulder).

    Navrae/Enquiries: René Kotzé
    Tel: 021 808 2653
    Email: renekotze@sun.ac.za

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