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We establish the existence of a stable foliation in the vicinity of atraveling front solution for systems of reaction diffusion equations in onespace dimension that arise in the study of chemical reactions models and solidfuel combustion. In this way we complement the orbital stability results fromearlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essentialspectrum of the differential operator obtained by linearization at the fronttouches the imaginary axis. In spaces with exponential weights, one can shiftthe spectrum to the left. We study the nonlinear equation on the intersectionof the unweighted and weighted spaces. Small translations of the front form acenter unstable manifold. For each small translation we prove the existence ofa stable manifold containing the translated front and show that the stablemanifolds foliate a small ball centered at the front.

Stable foliations near a traveling front for reaction diffusion systems