Abstract
In the paper, asymptotic solutions of the Cauchy problem with localized initial data for the two-dimensional wave equation (with variable speed) which is also perturbed by (spatially) variable weakly dispersive components are constructed. We consider both the case of normal dispersion occurring in the linearized Boussinesq equation for water waves over smoothly changing bottom and the case of anomalous dispersion arising when studying the wave equation with rapidly oscillating velocity. With regard to the fact that the front of the solution has focal points and self-intersection points, we present formulas based on the modified Maslov canonical operator in the case of initial perturbations of a rather general form which decrease at infinity. For perturbations of special form, we express the asymptotic behavior of a solution in the vicinity of the front, using derivatives of the sum of squares of the Airy functions Ai and Bi.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. N. Pelinovskii, Hydrodynamics of Tsunami Waves (Instutute of Applied Physics of the Russian Academy of Sciences, Nizhnii Novgorod, 1996) [in Russian].
C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientific, Singapore, 1989).
S. Yu. Dobrokhotov, S. Ya. Sekerzh-Zen’kovich, B. Tirozzi, and T. Ya. Tudorovskii, “Description of Tsunami Propagation Based on the Maslov Canonical Operator,” Dokl. Akad. Nauk. 409(2), 171–175 (2006) [Dokl. Math. 74 (1), 592–596 (2006)].
S. Yu. Dobrokhotov, B. Tirozzi, and C. A. Vargas, “Behavior near the Focal Points of Asymptotic Solutions to the Cauchy Problem for the Linearized Shallow Water Equations with Initial Lociclized Pertrubations,” Russ. J. Math. Phys. 16(2), 228–245 (2009).
S. Yu. Dobrokhotov, A. I. Shafarevich, and B. Tirozzi, “Localaized Wave and Vortical Solutions to Linear Hyperbolic Systems and Their Application to Linear Shallow Water Equations,” Russ. J. Math. Phys. 15(2), 192–221 (2008).
S. Yu. Dobrokhotov, R.V. Nekrasov, and B. Tirozzi, “Localaized Asymptotic Solutions of the Linear Shallow-Water Equations with Localized Initial Data,” J. Engrg. Math. 69(2), 225–242 (2011).
S. Yu. Dobrokhotov and P. N. Zhevandrov, “Asymptotic Expansions and the Maslov Canonical Operator in the Linear Theory of Water Waves. I. Main Constructions and Equations for Surface Gravity Waves,” Russ. J. Math. Phys. 10(1), 1–31 (2003).
D. Bianchi, S. Yu. Dobrokhotov, and B. Tirozzi, “Asymptotics of Localized Solutions of the One-Dimensional Wave Equation with Variable Velocity. II. Taking into Account the Right-Hand Side and a Weak Dispersion,” Russ. J. Math. Phys. 15(4), 427–446 (2008).
J. Bruening, V. V. Grushin, and S. Yu. Dobrokhotov, “Averaging of Linear Operators, Adiabatic Approximation, and Pseudodifferential Operators,” Mat. Zametki 92(2), 163–180 (2012) [Math. Notes 92 (2), 151–165 (2012)].
J. Bruening, V. V. Grushin, and S. Yu. Dobrokhotov, “Approximate Formulas for the Eigenvalues of a Laplace Operator, on a Torus, Which Arises in Linear Problems with Oscillating Coefficients,” Russ. J. Math. Phys. 19(3), 1–10 (2012).
S. Ya. Sekerzh-Zenkovich, “Simple Asymptotic Solution to the Cauchy-Poisson Problem for Leading Waves,” Russ. J. Math. Phys. 16(2), 215–222 (2009).
M. V. Berry, “Tsunami Asymptotics,” New J. Phys. 7(129), 1–18 (2005).
M. V. Berry, “Focused Tsunami Waves,” Proc. R. Soc. Lond. Ser. A 463, 3055–3071 (2007).
S. F. Dotsenko, B. Yu. Sergeevskii, and L. V. Cherkasov, “Space Tsunami Waves Generated by Alternating Displacement of the Ocean Surface,” Tsunami Research (1), 7–14 (1986) [in Russian].
S. Wang, “The Propagation of the Leading Wave,” ASCE Specialty Conf’ Coastal Hydrodynamics (University of Delaware, 1987), pp. 657–670.
S. Yu. Dobrokhotov, S. Ya. Sekerzh-Zenkovich, B. Tirozzi, and B. I. Volkov, “Asymptotic Description for Tsunami Waves in Framework of the Piston Model: General Construction and Explicitly Solvable Cases,” Fundam. Appl. Geophysics (SPb) (2), 15–29 (2009).
V. A. Borovikov, M. Ya. Kel’bert, “The Field near the Wave Front in a Cauchy-Poisson Problem,” Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza (2), 173–174 (1984) [Fluid Dynam. 19 (2), 321–323 (1984)].
V. P. Maslov, Operator Methods (Izdat. “Nauka”, Moscow, 1973) [in Russian].
M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization (Izdat. “Nauka”, Moscow, 1993; American Mathematical Society, Providence, RI, 1993).
V. Belov, S. Dobrokhotov, and T. Tudorovskiy, “Operator Separation of Variables for Adiabatic Problems in Quantum and Wave Mechanics,” J. Eng. Math. 55(1–4), 183–237 (2006).
J. J. Stoker, Water Waves (Interscience, New York, 1957).
V. P. Maslov and M. V. Fedoryuk, Quasiclassical Approximation for the Equations of Quantum Mechanics (Izdat. “Nauka”, Moscow, 1976) [in Russian].
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), 713–717 (2007)].
M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations (Izd. “Nauka,” Moscow, 1983; Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993).
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article is available at http://dx.doi.org/10.1134/S1061920815010161.
Rights and permissions
About this article
Cite this article
Dobrokhotov, S.Y., Sergeev, S.A. & Tirozzi, B. Asymptotic solutions of the Cauchy problem with localized initial conditions for linearized two-dimensional Boussinesq-type equations with variable coefficients. Russ. J. Math. Phys. 20, 155–171 (2013). https://doi.org/10.1134/S1061920813020040
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920813020040