  Consider a singularly perturbed system $$\epsilon u_t=\epsilon^2 u_{xx} +
f(u,x,\epsilon),\quad u\in {\Bbb R}^n,x\in{\Bbb R},t\geq 0. $$ 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 {\em 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 $\epsilon u_t=-(-\epsilon^2 D_{xx})^m u+ \dots.$
