Boundary-Value Problem for a System with Small Parameter

Abstract

In the previous chapter, we have considered some boundary-value problems for hyperbolic partial differential systems which can be reduced to difference, q-difference, differential-difference, and other functional or differential-functional equations. The reduction procedure makes it possible to use the highly developed theory of difference and differential-difference equations in order to investigate such boundary-value problems. In particular, one can use for this purpose the theory of discrete dynamical systems which is now being developed intensively. In many cases, this approach helps us to answer most of the questions we are interested in. However, the above-mentioned reduction procedure does not possess the property of “roughness” by itself, i.e., even for small regular perturbations of the original system an exact reduction becomes, as a rule, impossible. The following questions naturally arise: How does this feature affect the properties of the solutions which were described with its help ? Are these properties specific features of the reducible problems which disappear when the reduction becomes impossible? These questions are, clearly, very important from the physical point of view, since only properties stable with respect to perturbations have physical meaning.