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Lagrangian Manifolds and Efficient Short-Wave Asymptotics in a Neighborhood of a Caustic Cusp

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Abstract

We develop an approach to writing efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument. A constructive method is proposed that allows expressing the canonical operator near a caustic cusp corresponding to the Lagrangian singularity of type \(A_3\) (standard cusp) in terms of the Pearcey function and its first derivatives. It is shown that, conversely, the representation of a Pearcey type integral via the canonical operator turns out to be a very simple way to obtain its asymptotics for large real values of the arguments in terms of Airy functions and WKB-type functions.

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Notes

  1. A polynomial with unit leading coefficient.

  2. Of course, this is some liberty of speech; here and below, in similar statements, we mean that the function can be extended by continuity at the points where the denominator vanishes, and so the extended function is smooth.

  3. We choose the plus sign to be definite.

  4. We would like to take the point \((0,0)\) for the central point, but since this point is singular, we move away from it by the infinitesimal distance \(-\delta\) in the variable \(x_2\).

  5. Special charts with more general coordinates in [4] are not needed here.

  6. The union \(U_1\cup\Gamma_{1l}\cup U_0\cup\Gamma_{1r}\cup U_3\), naturally identified with \(\mathscr R\), is not a nonsingular chart, because its projection onto the \(x\)-plane is not one-to-one onto the image.

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Acknowledgments

The authors wish to express gratitude to A. V. Tsvetkova for valuable remarks.

Funding

This work was supported by the Russian Foundation for Basic Research under grant 17-01-00644.

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Authors and Affiliations

  1. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 119526, Moscow, Russia

    S. Yu. Dobrokhotov & V. E. Nazaikinskii

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  1. S. Yu. Dobrokhotov
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  2. V. E. Nazaikinskii
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Correspondence to S. Yu. Dobrokhotov.

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Dobrokhotov, S.Y., Nazaikinskii, V.E. Lagrangian Manifolds and Efficient Short-Wave Asymptotics in a Neighborhood of a Caustic Cusp. Math Notes 108, 318–338 (2020). https://doi.org/10.1134/S0001434620090023

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