Amplitude equations and asymptotic expansions for multi-scale problems

Article type: Research Article

Authors: Kirkinis, Eleftherios

Affiliations: Department of Applied Mathematics, University of Washington, Seattle, WA 98195-2420, USA. E-mail: kirkinis@amath.washington.edu

Abstract: In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases.

Keywords: perturbation methods, asymptotic analysis, boundary layers, nonlinear oscillators, partial differential equations

DOI: 10.3233/ASY-2009-0964

Journal: Asymptotic Analysis, vol. 67, no. 1-2, pp. 1-16, 2010