Hidden Dynamics and the Origin of Pulsating Waves in Self-propagating High Temperature Synthesis

We derive the precise limit of SHS in the high activation energy scaling suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J. Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit coincides with the Stefan problem for supercooled water with spatially inhomogeneous coefficients. In general it is a nonlinear forward-backward parabolic equation with discontinuous hysteresis term. In the first part of our paper we give a complete characterization of the limit problem in the case of one space dimension. Our precise form of the limit problem suggests a surprising explanation for the numerically observed pulsating waves. In the second part we construct a rather large family of pulsating waves for the limit problem.