Local and Global Existence of Multiple Waves Near Formal Approximations

Assuming that a formal approximation of multiple waves has been obtained by matched asymptotic methods, we derive a {\em Spatial Shadowing lemma} to construct exact solutions near the formal approximation. In Part I, we consider a general singularly perturbed parabolic system. $$\epsilon u_t + (-\epsilon^2)^m D^{2m}_x u = f(u,\epsilon u_x,\cdots,(\epsilon D_x)^{2m-1} u,x,\epsilon).$$ We show that if the formal approximation is precise, there is always an exact solution nearby for at least a short time. Examples include Cahn-Hilliard equation and viscous profile of conservation laws. In Part II, we show under some more assumptions, the process in Part I can be repeated to obtain global solutions if the formal approximation is a global one. Examples include reaction-diffusion equations and phase field equations.