Corner catastrophes and uniform asymptotics
type: Item
instance of: scholarly article
title: Corner catastrophes and uniform asymptotics
zbMATH Open document ID: 0879.35150
author: O. Diekmann
published in: Doklady Mathematics
publication date: 1998-01-18T00:00:00Z
review text: Asymptotic solutions to a number of partial differential equations are integrals of rapidly oscillating functions \[ U(\Lambda,\alpha)= \Lambda^{k/2}\underset G{}{\int\cdots\int} f(\eta,\alpha)\exp\{i\Lambda\Phi(\eta,\alpha)\}d\eta,\;\dim\alpha=m,\;\dim\eta=k,\tag{1} \] where \(\Lambda\gg 1\) is a parameter of asymptotic expansion, and \(f(\eta,\alpha)\) and \(\Phi(\eta,\alpha)\) are smooth functions. The degeneration of saddle points of the phase function \(\Phi\) in diffraction and wave propagation theory corresponds to caustics and their singularities of different types, and also corresponds to the focusing of the wave fields as well as to the formation of a complex diffraction structure. We generalize results of the theory of singularities of differentiable mappings (catastrophe theory) to ``corner catastrophes'' and apply our results to the analysis of integrals (1) with a domain of integration of the form \(G=[\eta^*_1,+\infty)\times [\eta^*_2,+\infty)\times \mathbb{R}^{k-2}\).
Mathematics Subject Classification ID: 35Q60
Mathematics Subject Classification ID: 78A45
Mathematics Subject Classification ID: 35C05
zbMATH DE Number: 1050637
zbMATH Keywords: integrals of rapidly oscillating functions
zbMATH Keywords: diffraction
zbMATH Keywords: caustics
zbMATH Keywords: singularities of differentiable mappings
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1050637