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 {\em with spatially inhomogeneous coefficients}. In general it is a nonlinear forward-backward parabolic equation {\em 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. In the second part we construct in any finite dimension a rather large family of pulsating waves for the limit problem. In the third part, we prove that for constant coefficients the limit problem in any finite dimension {\em does not admit non-trivial pulsating waves}. The combination of all three parts strongly suggests a relation between the pulsating waves constructed in the present paper and the numerically observed pulsating waves for finite activation energy in dimension $n\ge 1$ and therefore provides a possible and surprising explanation for the phenomena observed. All techniques in the present paper (with the exception of the remark in the Appendix) belong to the category far-from-equilibrium-analysis/far-from-bifurcation-point-analysis.