Hidden dynamics and the origin of pulsating waves in Self-propagating High temperature Synthesis

Details Hidden dynamics and the origin of pulsating waves in Self-propagating High temperature Synthesis Hidden dynamics and the origin of pulsating waves in Self-propagating High temperature Synthesis

2006-05-19

arxiv.org/abs/math/0605543v2

Mathematics

Analysis of PDEs

22 pages, 5 figures, the revised paper (version 2) is an extended version of the old paper

Scientific paper

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.

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