Standard methods of differential geometry are applied under different approaches: directly to reflector surfaces, and alternatively, to wavefronts, to obtain analytical caustics of two sheets or branches. Analytical caustics are obtained as a family of lines, a set of points, and several types of equations for geometries encountered in optics and microwave applications.
This document discusses the determination of caustic surfaces in terms of rays, reflectors, and wavefronts. As a by-product, these results also yield new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights, the calculation of correlation functions for certain The paper closes with a discussion of the relation of this work to recent developments in understanding the asymptotics of graphical enumeration.
Moreover, these insights point towards a more general program relating the first Painlevé hierarchy to the higher order structure of the double-scaling limit through the specific rational structure of generating functions in the genus expansion. This provides a precise expression of the Painlevé asymptotic coefficients directly in terms of the coefficients of the partial fractions expansion of the rational form of the generating functions established in this paper. As an intriguing application, one gains new insights into the relation between certain derivatives of the genus expansion, in a double-scaling limit, and the asymptotic expansion of the first Painlevé transcendent. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions. It is also a solution of the inviscid Burgers equation for certain initial data. This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of weight of the parameter t). The main results are to prove that these generating functions are, in fact, specific rational functions of a distinguished irrational (algebraic) function, z0(t). The coefficients of the large N expansion of the logarithm of this partition function, also known as the genus expansion (and its derivatives), are generating functions for a variety of graphical enumeration problems. This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. Caustics, counting maps and semi-classical asymptotics
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