Geodesic
The Maslov Canonical Operator on Lagrangian Manifolds in the Phase Space Corresponding to a~Wave Equation Degenerating on the Boundary
Matematičeskie zametki, Tome 96 (2014) no. 2, pp. 261-276.

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We construct the Maslov canonical operator on Lagrangian manifolds in the phase space corresponding to the wave equation in a domain on whose boundary the wave propagation velocity $c(x)$ degenerates as the square root of the distance from the boundary.
Keywords: wave equation, boundary, degeneration, asymptotics, Maslov canonical operator, Lagrangian manifold. 
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V. E. Nazaikinskii. The Maslov Canonical Operator on Lagrangian Manifolds in the Phase Space Corresponding to a~Wave Equation Degenerating on the Boundary. Matematičeskie zametki, Tome 96 (2014) no. 2, pp. 261-276. http://geodesic.mathdoc.fr/item/MZM_2014_96_2_a10/

[1] S. Yu. Dobrokhotov, B. Tirotstsi, “Lokalizovannye resheniya odnomernoi nelineinoi sistemy uravnenii melkoi vody so skorostyu $c=\sqrt x$”, UMN, 65:1 (2010), 185–186 | DOI | MR | Zbl

[2] S. Yu. Dobrokhotov, V. E. Nazaikinskii, B. Tirozzi, “Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity. I”, Russ. J. Math. Phys., 17:4 (2010), 434–447 | DOI | MR | Zbl

[3] 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

[4] V. E. Nazaikinskii, “Asimptoticheskie resheniya vyrozhdayuschegosya volnovogo uravneniya s lokalizovannymi nachalnymi dannymi, otvechayuschie razlichnym samosopryazhennym rasshireniyam”, Matem. zametki, 89:5 (2011), 797–800 | DOI | MR | Zbl

[5] V. E. Nazaikinskii, “Geometriya fazovogo prostranstva dlya volnovogo uravneniya, vyrozhdayuschegosya na granitse oblasti”, Matem. zametki, 92:1 (2012), 153–156 | DOI | Zbl

[6] 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

[7] J. J. Stoker, Water Waves: The Mathematical Theory with Applications, John Wiley Sons, New York, 1992 | MR | Zbl

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

[9] O. A. Oleinik, E. V. Radkevich, “Uravneniya vtorogo poryadka s neotritsatelnoi kharakteristicheskoi formoi”, Itogi nauki. Ser. Matematika. Mat. anal. 1969, VINITI, M., 1971, 7–252 | MR | Zbl

[10] G. Fichera, “Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine”, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. (8), 5 (1956), 3–30 | MR | Zbl

[11] M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, LGU, L., 1980 | MR

[12] T. Vukašinac, P. Zhevandrov, “Geometric asymptotics for a degenerate hyperbolic equation”, Russ. J. Math. Phys., 9:3 (2002), 371–381 | MR | Zbl

[13] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, MGU, M., 1965

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

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

[16] V. E. Nazaikinskii, “O predstavleniyakh lokalizovannykh funktsii v $\mathbb{R}^2$ kanonicheskim operatorom Maslova”, Matem. zametki, 96:1 (2014), 87–99 | DOI

[17] V. S. Vladimirov, Uravneniya matematicheskoi fiziki, Nauka, M., 1981 | MR

[18] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR | Zbl

[19] Funktsionalnyi analiz, ed. S. G. Krein, Nauka, M., 1972 | MR | Zbl

[20] V. P. Maslov, Operatornye metody, Nauka, M., 1973 | MR | Zbl

[21] L. Khërmander, Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi. T. 3. Psevdodifferentsialnye operatory, Mir, M., 1987 | MR

[22] G. Beitmen, A. Erdeii, Tablitsy integralnykh preobrazovanii. T. 2. Preobrazovaniya Besselya. Integraly ot spetsialnykh funktsii, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1970

[23] G. N. Vatson, Teoriya besselevykh funktsii. Ch. 1, IL, M., 1949

[24] A. S. Mischenko, B. Yu. Sternin, V. E. Shatalov, Lagranzhevy mnogoobraziya i metod kanonicheskogo operatora, Nauka, M., 1978 | MR

[25] V. I. Arnold, “O kharakteristicheskom klasse, vkhodyaschem v usloviya kvantovaniya”, Funkts. analiz i ego pril., 1:1 (1967), 1–14 | MR | Zbl

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