Asymptotics of the Helmholtz equation solutions in a two-layer medium with a localized right-hand side

A. Yu. Anikin ; A. I. Klevin

Teoretičeskaâ i matematičeskaâ fizika, Tome 216 (2023) no. 1, pp. 148-168

Résumé

We construct the leading term of the semiclassical asymptotic solution of the Helmholtz equation with a small parameter in the localized right-hand side. This equation arises, for example, in the problem of ocean acoustics, in which the small parameter is given by the ratio of the characteristic scale of the “vertical” coordinate to that of the other coordinates. The equation is considered in the region bounded in the “vertical” coordinate; it is divided into two layers, with the coefficient in the Helmholtz equation and the derivative of the solution having fixed jump discontinuities at the interface. The technique for constructing the asymptotics involves the operator separation of variables (adiabatic approximation) and the use of the recently developed method for constructing asymptotics of equations with localized right-hand sides in the equations obtained after the variable separation.

Détail

Keywords: Helmholtz equation, equation with a right-hand side, semiclassical asymptotics, Maslov canonical operator.

Bibliographie

[1] B. Katsnelson, V. Petnikov, J. Lynch, Fundamentals of Shallow Water Acoustics, Springer, New York, 2012 | DOI

[2] F. B. Jensen, W. A. Kuperman, M. B. Porter, H. Schmidt, Computational Ocean Acoustics, Springer, New York, 2011 | DOI | MR

[3] G. V. Frisk, Ocean and Seabed Acoustics: A Theory of Wave Propagation, Prentice Hall PTR, Englewood Cliffs, NJ, 1998

[4] C. L. Pekeris, “Theory of propagation of explosive sound in shallow water”, Propagation of Sound in the Ocean, Geological Society of America Memoirs, 27, eds. J. L. Worzel; M. Ewing, C. L. Pekeris, Geol. Soc. Amer., New York, 1948, 1–118 | DOI

[5] P. S. Petrov, M. Yu. Trofimov, A. D. Zakharenko, “Modal perturbation theory in the case of bathymetry variations in shallow-water acoustics”, Russ. J. Math. Phys., 28:2 (2021), 257–262 | DOI | MR

[6] A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rulo, “Lagranzhevy mnogoobraziya i konstruktsiya asimptotik dlya (psevdo)differentsialnykh uravnenii s lokalizovannymi pravymi chastyami”, TMF, 214:1 (2023), 3–29 | DOI | DOI

[7] V. M. Babich, “O korotkovolnovoi asimptotike funktsii Grina dlya uravneniya Gelmgoltsa”, Matem. sb., 65(107):4 (1964), 576–630 | MR | Zbl

[8] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln, Nauka, M., 1972 | MR

[9] V. V. Belov, S. Yu. Dobrokhotov, V. P. Maslov, T. Ya. Tudorovskii, “Obobschennyi adiabaticheskii printsip dlya opisaniya dinamiki elektrona v iskrivlennykh nanostrukturakh”, UFN, 175:9 (2005), 1004–1010 | DOI | DOI

[10] V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskiy, “Operator separation of variables for adiabatic problems in quantum and wave mechanics”, J. Eng. Math., 55:1–4 (2006), 183–237 | DOI | MR

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

[12] R. P. Feynman, “An operator calculus having applications in quantum electrodynamics”, Phys. Rev., 84:1 (1951), 108–128 | DOI | MR

[13] A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, M. Rulo, “Kanonicheskii operator Maslova na pare lagranzhevykh mnogoobrazii i asimptotika reshenii statsionarnykh uravnenii s lokalizovannymi pravymi chastyami”, Dokl. RAN, 475:6 (2017), 624–628 | DOI | Zbl

[14] S. Yu. Dobrokhotov, D. S. Minenkov, M. Rulo, “Printsip Mopertyui–Yakobi dlya gamiltonianov vida $f(x,|p|)$ v nekotorykh dvumernykh statsionarnykh kvaziklassicheskikh zadachakh”, Matem. zametki, 97:1 (2015), 48–57 | DOI | DOI | MR | Zbl

[15] S. Yu. Dobrokhotov, M. Rulo, “Kvaziklassicheskii analog printsipa Mopertyui–Yakobi i ego prilozhenie k lineinoi teorii voln na vode”, Matem. zametki, 87:3 (2010), 458–463 | DOI | DOI | MR | Zbl

[16] S. Yu. Dobrokhotov, M. Rouleux, “The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic hamiltonian flows with applications to linear water waves theory”, Asymptotic Anal., 74:1–2 (2011), 33–73 | DOI | MR

[17] A. Yu. Anikin, S. Yu. Dobrokhotov, A. I. Klevin, B. Tirotstsi, “Skalyarizatsiya statsionarnykh kvaziklassicheskikh zadach dlya sistem uravnenii i prilozhenie k fizike plazmy”, TMF, 193:3 (2017), 409–433 | DOI | DOI

[18] B. R. Vainberg, “O korotkovolnovoi asimptotike reshenii statsionarnykh zadach i asimptotike pri $t\to\infty$ reshenii nestatsionarnykh zadach”, UMN, 30:2(182) (1975), 3–55 | DOI | MR | Zbl

[19] P. N. Petrov, S. Yu. Dobrokhotov, “Asimptotika resheniya uravneniya Gelmgoltsa v trekhmernom sloe peremennoi tolschiny s lokalizovannoi pravoi chastyu”, Zh. vychisl. matem. i matem. fiz., 59:4 (2019), 566–578 | DOI | DOI | MR | Zbl

[20] V. P. Maslov, Teoriya vozmuschenii i asimptoticheskie metody, Izd-vo Mosk. un-ta, M., 1965 | Zbl

[21] G. Teschl, Mathematical Methods in Quantum Mechanics, Graduate Studies in Mathematics, 157, AMS, Providence, RI, 2014 | DOI | MR

[22] A. Zettl, Sturm–Liouville Theory, Mathematical Surveys and Monographs, 121, AMS, Providence, RI, 2005 | DOI | MR

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

[24] S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. I. Shafarevich, “Novye integralnye predstavleniya kanonicheskogo operatora Maslova v osobykh kartakh”, Izv. RAN. Ser. matem., 81:2 (2017), 53–96 | DOI | DOI | MR

[25] S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Efficient formulas for the Maslov canonical operator near a simple caustic”, Russ. J. Math. Phys., 25:4 (2018), 545–552 | DOI | MR

[26] S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Lagranzhevy mnogoobraziya i effektivnye formuly dlya korotkovolnovykh asimptotik v okrestnosti tochki vozvrata kaustiki”, Matem. zametki, 108:3 (2020), 334–359 | DOI | DOI | MR