hyperbolic umbilic catastrophe

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11: 36.12 Uniform Approximation of Integrals

… ►Define a mapping $u(t;𝐲)$ by relating $f(u;𝐲)$ to the normal form (36.2.1) of ${\mathrm{\Phi }}_K(t;𝐱)$ in the following way: …with the $K+1$ functions $A(𝐲)$ and $𝐱(𝐲)$ determined by correspondence of the $K+1$ critical points of $f$ and ${\mathrm{\Phi }}_K$. … ►This technique can be applied to generate a hierarchy of approximations for the diffraction catastrophes ${\mathrm{\Psi }}_K(𝐱;k)$ in (36.2.10) away from $𝐱=\mathrm{𝟎}$, in terms of canonical integrals ${\mathrm{\Psi }}_J\left(\xi (𝐱;k)\right)$ for . For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. … ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).