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Dissipationless shocks and Painleve equations

Presented by: 
C Klein [Leipzig]
Tuesday 19th September 2006 - 11:30 to 12:00
INI Seminar Room 1

The Cauchy problem for dissipationless equations as the Korteweg de Vries (KdV) equation with small dispersion of order $\epsilon^2$, $\epsilon\ll 1$, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order $\epsilon$. Near the gradient catastrophe of the dispersionless equation ($\epsilon=0$), a multi-scales expansion gives an asymptotic solution in terms of a fourth order generalization of Painlev\'e I. At the leading edge of the oscillatory zone, a corresponding multi-scales expansion yields an asymptotic description of the oscillations where the envelope is given by a solution to the Painlev\'e II equation. We study the applicability of these approximations for several PDEs and random matrix models numerically.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons