  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.
