The application of centre-manifold theory to the evolution of system which vary slowly in space

Abstract

In many physical problems, the system tends quickly to a particular structure, which then evolves relatively slowly in space and time. Various methods exist to derive equations describing the slow evolution of the particular structure; for example, the method of multiple scales. However, the resulting equations are typically valid only for a limited range of the parameters. In order to extend the range of validity and to improve the accuracy, correction terms must be found for the equations. Here we describe a procedure, inspired by centre-manifold theory, which provides a systematic approach to calculating a sequence of successively more accurate approximations to the evolution of the principal structure in space and time.The formal procedure described here raises a number of questions for future research. For example: what sort of error bounds can be obtained, do the approximations converge or are they strictly asymptotic, and what sort of boundary conditions are appropriate in a given problem?