Geodesic
On the Representation of Localized Functions in~$\mathbb R^2$ by Maslov's Canonical Operator
Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 88-100.

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We prove that localized functions can be represented in the form of an integral over a parameter, the integrand being Maslov's canonical operator applied to an amplitude obtained from the Fourier transform of the function to be represented. This representation generalizes an earlier one obtained by Dobrokhotov, Tirozzi, and Shafarevich and permits representing localized initial data for wave type equations with the use of an invariant Lagrangian manifold, which simplifies the asymptotic solution formulas dramatically in many cases.
Keywords: wave equation, asymptotics, localized initial data, integral representation, invariant Lagrangian manifold, Maslov's canonical operator. 
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V. E. Nazaikinskii. On the Representation of Localized Functions in~$\mathbb R^2$ by Maslov's Canonical Operator. Matematičeskie zametki, Tome 96 (2014) no. 1, pp. 88-100. http://geodesic.mathdoc.fr/item/MZM_2014_96_1_a7/

[1] V. P. Maslov, M. V. Fedoryuk, Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR | Zbl

[2] S. Yu. Dobrokhotov, B. Tirotstsi, A. I. Shafarevich, “Predstavleniya bystroubyvayuschikh funktsii kanonicheskim operatorom Maslova”, Matem. zametki, 82:5 (2007), 792–796 | DOI | MR | Zbl

[3] S. Yu. Dobrokhotov, P. N. Zhevandrov, V. P. Maslov, A. I. Shafarevich, “Asimptoticheskie bystroubyvayuschie resheniya lineinykh strogo giperbolicheskikh sistem s peremennymi koeffitsientami”, Matem. zametki, 49:4 (1991), 31–46 | MR | Zbl

[4] S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, T. Tudorovski, “Asymptotic theory of tsunami waves: geometrical aspects and the generalized Maslov representation”, Publications of the Research Institute for Mathematical Sciences, 4:1482 (2006), 118–153

[5] S. Yu. Dobrokhotov, S. Ya. Sekerzh-Zenkovich, B. Tirotstsi, T. Ya. Tudorovskii, “Opisanie rasprostraneniya voln tsunami na osnove kanonicheskogo operatora Maslova”, Dokl. RAN, 409:2 (2006), 171–175 | MR

[6] S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, B. Volkov, “Explicit asymptotics for tsunami waves in framework of the piston model”, Russ. J. Earth Sci., 8 (2006), ES4003 | DOI

[7] S. Yu. Dobrokhotov, A. I. Shafarevich, B. Tirozzi, “Localized wave and vortical solutions of linear hyperbolic systems and their application to the linear shallow water equations”, Russ. J. Math. Phys., 15:2 (2008), 192–221 | DOI | MR | Zbl

[8] S. Yu. Dobrokhotov, B. Tirozzi, C. A. Vargas, “Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations”, Russ. J. Math. Phys., 16:2 (2009), 228–245 | DOI | MR | Zbl

[9] V. I. Arnold, Osobennosti kaustik i volnovykh frontov, Biblioteka matematika, 1, Fazis, M., 1996 | MR

[10] S. Yu. Dobrokhotov, G. N.Makrakis, V. E. Nazaikinskii, T. Ya. Tudorovskii, “Novye formuly dlya kanonicheskogo operatora Maslova v okrestnosti fokalnykh tochek i kaustik v dvumernykh kvaziklassicheskikh asimptotikakh”, TMF, 177:3 (2013), 355–386 | DOI

[11] S. Yu. Dobrokhotov, S. O. Sinitsyn, B. Tirozzi, “Asymptotics of localized solutions of the one-dimensional wave equation with variable velocity. I. The Cauchy problem”, Russ. J. Math. Phys., 14:1 (2007), 28–56 | DOI | MR | Zbl

[12] S. Yu. Dobrokhotov, V. E. Nazaikinskii, B. Tirotstsi, “Asimptoticheskie resheniya dvumernogo modelnogo volnovogo uravneniya s vyrozhdayuscheisya skorostyu i lokalizovannymi nachalnymi dannymi”, Algebra i analiz, 22:6 (2010), 67–90 | MR | Zbl

[13] S. Yu. Dobrokhotov, V. E. Nazaikinskii, B. Tirozzi, “Two-dimensional wave equation with degeneration on the curvilinear boundary of the domain and asymptotic solutions with localized initial data”, Russ. J. Math. Phys, 20:4 (2013), 389–401 | DOI | MR

[14] L. Hörmander, “Fourier integral operators. I”, Acta Math., 127 (1971), 79–183 | DOI | MR | Zbl

[15] M. V. Fedoryuk, Metod perevala, Nauka, M., 1977 | MR

[16] S. F. Dotsenko, B. Yu. Sergievskii, L. V. Cherkasov, “Prostranstvennye volny tsunami, vyzvannye znakoperemennym smescheniem poverkhnosti okeana”, Issledovaniya tsunami, 1986, no. 1, 7–14

[17] S. Wang, “The propagation of the leading wave”, Coastal Hydrodynamics, ASCE Specialty Conference on Coastal Hydrodynamics (June 28–July 1, University of Delaware), American Society of Civil Engineers, New York, NY, 1987, 657–670

[18] E. N. Pelinovskii, Gidrodinamika voln tsunami, IPF RAN, Nizhnii Novgorod, 1996

[19] S. Ya. Sekerzh-Zenkovich,, “Simple asymptotic solution to the Cauchy–Poisson problem for leading waves”, Russ. J. Math. Phys., 16:2 (2009), 315–322 | DOI | MR | Zbl

[20] S. Yu. Dobrokhotov, S. Ya. Sekerzh-Zenkovich, “Odin klass tochnykh algebraicheskikh lokalizovannykh reshenii mnogomernogo volnovogo uravneniya”, Matem. zametki, 88:6 (2010), 942–945 | DOI | MR | Zbl