Shadowing Matching Errors for

Consider a singularly perturbed system =ut==2uxx+ f(u, x, =), u # Rn, x # R, t0. Assume that the system has a sequence of regular and internal layers occurring alternatively along the x direction. These ‘‘multiple wave’ ’ solutions can formally be constructed by matched asymptotic expansions. To obtain a genuine solution, we derive a Spatial Shadowing Lemma which assures the existence of an exact solution that is near the formal asymptotic series provided (1) the residual errors are small in all the layers, and (2) the matching errors are small along the lateral boundaries of the adjacent layers. The method should work on some other systems like =ut=&(&=2Dxx)m u+} } }. 1996 Academic Press, Inc. 1