In the approximation of linearized shallow water equations, the free surface elevation for gravitational waves in a basin Ω of variable depth D(x) with gently sloping beaches is governed by a divergence form wave equation with the squared velocity c2 = gD(x) degenerating on ∂Ω = {x ∈ R2 : D = 0}. It is assumed that ∇D ≠ 0 for x ∈ ∂Ω. We consider a class of problems on manifolds with boundary generalizing this example. The phase space for such problems is the symplectic reduction of the cotangent bundle of a closed manifold of dimension higher by one. We construct semiclassical asymptotics for equations under consideration by using the quantization of reduction procedure applied to the Maslov canonical operator on Lagrangian submanifolds of the closed manifold.
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References
V. E. Nazaikinskii, “On an elliptic operator degenerating on the boundary” [in Russian], Funkts. Anal. Prilozh. 56, No. 4, 109–112 (2022); English transl.:Funct. Anal. Appl. 56, No. 4, 324–326 (2022).
V. P. Maslov, Téorie des Perturbations et Méthodes Asymptotiques, Dunod, Paris (1972).
V. P. Maslov and M. V. Fedoryuk, Semi-Cclasical Approximation in Quantum Mechanics, D. Reidel, Dordrecht etc. (1981).
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and A. I. Shafarevich, “New integral representations of the Maslov canonical operator in singular charts,” Izv. Math. 81, No. 2, 286–328 (2017).
V. E. Nazaikinskii, “The Maslov canonical operator on Lagrangian manifolds in the phase space corresponding to a wave equation degenerating on the boundary,” Math. Notes 96, No. 2, 248–260 (2014).
A. Yu. Anikin, S. Yu. Dobrokhotov, and V. E. Nazaikinskii, “Simple asymptotics for a wave equation with degenerating velocity and their applications in the linear long wave run-up problem,” Math. Notes 104, No. 4, 471–488 (2018).
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and A. A. Tolchennikov, “Uniform asymptotics of the boundary values of the solution in a linear problem on the run-up waves on a shallow beach,” Math. Notes 101, No. 5, 802–814 (2017).
J. Marsden and A. Weinstein, “Reduction of symplectic manifolds with symmetry,” Rep. Math. Phys. 5, No. 1, 121–130 (1974).
S. Yu. Dobrokhotov and V. E. Nazaikinskii, “Uniformization of equations with Bessel-type boundary degeneration and semiclassical asymptotics,” Math. Notes 107, No. 5, 847–853 (2020).
J. Lott, “Signatures and higher signatures of 𝕊1-quotients,” Math. Ann. 316, No. 4, 617–657 (2000).
V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, Berlin (1997).
S. Yu. Dobrokhotov and V. E. Nazaikinskii, “On the asymptotics of a Bessel-type integral having applications in wave run-up theory,” Math. Notes 102, No. 6, 756–762 (2017).
O. A. Oleinik and E. V. Radkevich, “Second order equations with nonnegative characteristic form” [in Russian], Itogi Nauki, Ser. Mat., Mat. Anal. 1969, 7–252 (1971).
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Translated from Problemy Matematicheskogo Analiza 122, 2023, pp. 5-24.
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Anikin, A.Y., Dobrokhotov, S.Y., Nazaikinskii, V.E. et al. Uniformization and Semiclassical Asymptotics for a Class of Equations Degenerating on the Boundary of a Manifold. J Math Sci 270, 507–530 (2023). https://doi.org/10.1007/s10958-023-06363-8
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DOI: https://doi.org/10.1007/s10958-023-06363-8