Abstract
We consider the Cauchy problem with spatially localized initial data for the twodimensional wave equation degenerating on the boundary of the domain. This problem arises, in particular, in the theory of tsunami wave run-up on a shallow beach. Earlier, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi developed a method for constructing asymptotic solutions of this problem. The method is based on a modified Maslov canonical operator and on characteristics (trajectories) unbounded in the momentum variables; such characteristics are nonstandard from the viewpoint of the theory of partial differential equations. In a neighborhood of the velocity degeneration line, which is a caustic of a special form, the canonical operator is defined via the Hankel transform, which arises when applying Fock’s quantization procedure to the canonical transformation regularizing the above-mentioned nonstandard characteristics in a neighborhood of the velocity degeneration line (the boundary of the domain). It is shown in the present paper that the restriction of the asymptotic solutions to the boundary is determined by the standard canonical operator, which simplifies the asymptotic formulas for the solution on the boundary dramatically; for initial perturbations of special form, the solutions can be expressed via simple algebraic functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. S. Vladimirov, The Equations ofMathematical Physics (Nauka, Moscow, 1988) [in Russian].
M. Sh. Birman and M. Z. Solomyak, “Spectral Theory of Self-Adjoint Operators in Hilbert Space,” (Izd. Leningrad. Univ., Leningrad, 1980) [in Russian].
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and 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), 434–447 (2010).
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi, “Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data,” Algebra Anal. 22 (6), 67–90 (2010) [St. Petersburg Math. J. 22 (6), 895–911 (2011)].
S. Yu. Dobrokhotov, V. E. Nazaikinskii, and 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), 389–401 (2013).
J. J. Stoker, Water Waves. The Mathematical Theory with Applications (John Wiley & Sons, New York, 1992).
E. N. Pelinovskii, Hydrodynamics of Tsunami Waves (Institute of Applied Physics, Russian Academy of Sciences, Nizhni Novgorod, 1996) [in Russian].
S. Yu. Dobrokhotov and V. E. Nazaikinskii, “Asymptotics of wave and vortical localized solutions of the linearized shallow water equations,” in Topical Problems inMechanics (Nauka, Moscow, 2015), pp. 98–139 [in Russian].
S. I. Kabanikhin and O. I. Krivorot’ko, “An algorithm for computing wavefront amplitudes and inverse problems (tsunami, electrodynamics, acoustics, and viscoelasticity),” Dokl. Ross. Akad. Nauk 466 (6), 645–649 (2016) [Dokl. Math. 93 (1), 103–107 (2016)].
S. I. Kabanikhin and O. I. Krivorot’ko, “A numerical algorithm for computing tsunami wave amplitude,” Sibirsk. Zh. Vychisl. Mat. 19 (2), 153–165 (2016).
V. E. Nazaikinskii, “Phase space geometry for a wave equation degenerating on the boundary of the domain,” Mat. Zametki 92 (1), 153–156 (2012) [Math. Notes 92 (1–2), 144–148 (2012)].
T. Vukašinac and P. Zhevandrov, “Geometric asymptotics for a degenerate hyperbolic equation,” Russ. J. Math. Phys. 9 (3), 371–381 (2002).
V. P. Maslov, Perturbation Theory and Asymptotic Methods (Izd. Moskov. Univ., Moscow, 1965) [in Russian].
V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation for the Equations of Quantum Mechanics (Nauka, Moscow, 1976) [in Russian].
V. A. Fock, “On the canonical transformation in classical and quantum mechanics,” Vestn. Leningrad. Univ. 16, 67–70 (1959) [Acta Phys. Acad. Sci. Hungaricae 27 (1–4), 219–224 (1969)].
G. F. Carrier and H. P. Greenspan, “Water waves of finite amplitude on a sloping beach,” J. Fluid Mech. 4 (1), 97–109 (1958).
S. Yu. Dobrokhotov, B. Tirozzi, and A. I. Shafarevich, “Representations of rapidly decaying functions by the Maslov canonical operator,” Mat. Zametki 82 (5), 792–796 (2007) [Math. Notes 82 (5–6), 713–717 (2007)].
V. E. Nazaikinskii, “On the representations of localized functions in R2 by the Maslov canonical operator,” Mat. Zametki 96 (1), 88–100 (2014) [Math. Notes 96 (1–2), 99–109 (2014)].
S. Ya. Sekerzh-Zen’kovich, “Simple asymptotic solution of the Cauchy–Poisson problem for head waves,” Russ. J. Math. Phys. 16 (2), 315–322 (2009).
V. E. Nazaikinskii, “TheMaslov canonical operator on Lagrangian manifolds in the phase space corresponding to a wave equation degenerating on the boundary,” Mat. Zametki 96 (2), 261–276 (2014) [Math. Notes 96 (1–2), 248–260 (2014)].
S. Yu. Dobrokhotov, G. N. Makrakis, V. E. Nazaikinskii, and T. Ya. Tudorovskii, “New formulas for Maslov’s canonical operator in a neighborhood of focal points and caustics in two-dimensional semiclassical asymptotics,” Teoret. Mat. Fiz. 177 (3), 355–386 (2013) [Theoret. and Math. Phys. 177 (3), 1579–1605 (2013)].
V. I. Arnol’d, “Characteristic class entering in quantization conditions,” Funktsional. Anal. Prilozhen 1 (1), 1–14 (1967) [Functional Anal. Appl. 1 (1), 1–13 (1967)].
V. I. Arnol’d, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1974) [in Russian].
H. Bateman and A. Erdélyi, Tables of Integral Transforms (McGraw-Hill, New York–Toronto–London, 1954; Nauka, Moscow, 1970), Vol. 2.
M. V. Fedoryuk, Mountain Pass Method (Nauka, Moscow, 1977) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © S. Yu. Dobrokhotov, V. E. Nazaikinskii, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 5, pp. 710–731.
Rights and permissions
About this article
Cite this article
Dobrokhotov, S.Y., Nazaikinskii, V.E. Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation. Math Notes 100, 695–713 (2016). https://doi.org/10.1134/S0001434616110067
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434616110067