## Abstract

A theory is developed for the derivation of formal asymptotic solutions for initial boundary-value problems for equations of the form A$^{0}\frac{\partial {\bf u}}{\partial t}$ + $\underset \nu =1\to{\overset n\to{\sum}}$A$^{\nu}\frac{\partial {\bf u}}{\partial x_{\nu}}$ + $\lambda $Bu + Cu = f(t, X; $\lambda $), where A$^{0}$, A$^{\nu}$, B, and C are m $\times $ m matrix functions of t and X = (x$_{1}$, $\ldots $, x$_{n}$), u(t, X; $\lambda $) is an m-component column vector, and $\lambda $ is a large positive parameter. Our procedure is to consider a formal asymptotic solution of the form u(t, X; $\lambda $) $\sim $ e$^{^{1}\lambda s(t,{\bf x})}\underset j=0\to{\overset \infty \to{\sum}}$ (i$\lambda $)$^{-j}$ z$_{j}$(t, X). Substitution of this formal solution into the equation yields, for the function s(t, X), a first order partial differential equation which can be solved by the method of characteristics. If the coefficient matrices satisfy certain conditions then we obtain, for the functions z$_{j}$(t, X), linear systems of ordinary differential equations called transport equations along space-time curves called rays. They may be solved explicitly under suitable conditions. A proof is presented of the asymptotic nature of the formal solution when the coefficient matrices and initial data for u are appropriately chosen. The problem of reflexion and refraction at an interface is considered.