What distinguishes order from chaos? How can we identify systems that are integrable and only have ordered solutions?
There is strong evidence that the answer depends on the complex analytic properties of the systems of interest. The Painlev\'e equations (six classical nonlinear ordinary differential equations) are prototypical examples of this evidence. Much of my research concentrates on these deep and beautiful equations.
These equations were named after Paul Painlev\'e, a French mathematician and aviation enthusiast who was also a prime minister (twice) early in this century. The central idea that links complex analyticity and integrability was initiated by Sofia Kovalevskaya, one of my mathematical heroes.
Amazingly, the above ideas on integrable differential equations also extend to difference equations, and even to extended versions of cellular automata. While opinions differ on this topic, I find the beautiful constructions of automata by Stephen Wolfram very seductive.
Asymptotics is closely related to the techniques used for finding singularity structure. Pierre Boutroux initiated a deep and extensive asymptotic study of the Painlev\'e equations. George Gabriel Stokes was an amazing mathematical physicist who initiated many profound asymptotic studies, some of which are only now being extended to nonlinear equations.
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Last modified: 26 July 2011 by N.Joshi