Corner catastrophes and uniform asymptotics
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 \(G=[\eta^*_1,+\infty)\times [\eta^*_2,+\infty)\times \mathbb{R}^{k-2}\).
integrals of rapidly oscillating functions
singularities of differentiable mappings
scientific article; zbMATH DE number 1050637