Professor Triantafyllos Akylas on: Exponential asymptotics: introduction and model problems
Abstract: The classical singular perturbation methods of matched asymptotic expansions and multiple scales are powerful analytical tools in science and engineering. In spite of numerous successes, however, these techniques also suffer limitations. One notable shortcoming manifests in problems where the physics of interest happens to lie ‘beyond all orders’ and thus cannot be captured by expansions in powers of a small parameter. This difficulty arises, for instance, at the tails of gravity-capillary solitary waves in shallow water, which feature short-scale (capillary) oscillations of exponentially small amplitude relative to the solitary wave core. A similar issue is also encountered at the tails of internal gravity solitary waves in a stratified fluid layer. Furthermore, exponentially small effects play an important part in various other contexts, including envelope solitary waves on deep water, gap solitons in periodic media, free-surface flow past submerged bodies at low Froude number, pattern formation and crystal growth.
The purpose of exponential asymptotics is to carry standard perturbation expansions beyond all orders and thereby reveal the presence of exponentially small terms. To this end, two separate methods of attack have been developed. First, a WKB approach that focuses on singularities of the standard expansions in the complex plane where these expansions become disordered, and collects the desired exponentially small terms either (i) by matching with inner expansions valid close to the singularities; or (ii) by examining the switching on of exponentially small terms across Stokes lines. The second approach is specifically tailored to computing exponentially small short-scale waves. Such disturbances are tied to simple poles on the real axis in the Fourier (wavenumber) domain so the analysis focuses on determining the residues of these poles, which have exponentially small magnitude. This task is accomplished using standard multiple-scale and matched-asymptotics techniques.
This Lecture will focus on the exponential asymptotics procedure in the wavenumber domain. Specifically, for simplicity, the salient aspects of this approach will be discussed in the context of two model problems: (i) solitary waves of the Korteweg-deVries (KdV) equation with a fifth-order derivative term; (ii) the downstream wave response of the steady forced KdV equation with a smooth locally confined forcing term and the radiation condition of no waves far upstream.