Asymptotic solutions of the two-dimensional model wave equation with degenerating velocity and localized initial data

Authors:
S. Yu. Dobrokhotov, V. E. Nazaĭkinskiĭ and B. Tirozzi

Translated by:
the authors

Original publication:
Algebra i Analiz, tom **22** (2010), nomer 6.

Journal:
St. Petersburg Math. J. **22** (2011), 895-911

MSC (2010):
Primary 35L05

DOI:
https://doi.org/10.1090/S1061-0022-2011-01175-6

Published electronically:
August 18, 2011

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Cauchy problem is considered for the two-dimensional wave equation with velocity on the half-plane , with initial data localized in a neighborhood of the point . This problem serves as a model problem in the theory of beach run-up of long small-amplitude surface waves excited by a spatially localized instantaneous source. The asymptotic expansion of the solution is constructed with respect to a small parameter equal to the ratio of the source linear size to the distance from the -axis (the shoreline). The construction involves Maslov's canonical operator modified to cover the case of localized initial conditions. The relationship of the solution with the geometrical optics ray diagram corresponding to the problem is analyzed. The behavior of the solution near the -axis is studied. Simple solution formulas are written out for special initial data.

**1.**C. C. Mei,*The applied dynamics of ocean surface waves*, World Sci., Singapore, 1989.**2.**J. J. Stoker,*Water waves: The mathematical theory with applications*, Pure Appl. Math., vol. 4, Intersci. Publ., Inc., New York, 1957 (reprinted in 1992). MR**0103672 (21:2438)**; MR**1153414 (92m:76029)****3.**E. N. Pelinovskiĭ,*Hydrodynamics of tsunami waves*, Inst. Prikl. Fiz. Ross. Akad. Nauk, Nizhniĭ Novgorod, 1996. (Russian)**4.**S. Dobrokhotov, A. Shafarevich, and B. Tirozzi,*Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations*, Russ. J. Math. Phys.**15**(2008), no. 2, 192-221. MR**2410830 (2009c:35271)****5.**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 localized perturbations*, Russ. J. Math. Phys.**16**(2009), no. 2, 228-245. MR**2525408 (2010m:35210)****6.**V. P. Maslov,*Perturbation theory and asymptotic methods*, Moskov. Gos. Univ., Moscow, 1965. (Russian)**7.**S. Yu. Dobrokhotov, B. Tirozzi, and A. I. Shafarevich,*Representations of rapidly decreasing functions by the Maslov canonical operator*, Mat. Zametki**82**(2007), no. 5, 792-796; English transl., Math. Notes**82**(2007), no. 5-6, 713-717. MR**2399958 (2009d:58051)****8.**T. Vukašinac and P. Zhevandrov,*Geometric asymptotics for a degenerate hyperbolic equation*, Russ. J. Math. Phys.**9**(2002), no. 3, 371-381. MR**1965389 (2004d:35180)****9.**V. A. Borovikov,*Uniform stationary phase method*, IEE Electromagnetic Waves Ser., vol. 40, IEE, London, 1994. MR**1325462 (96b:58111)****10.**M. V. Fedoryuk,*The saddle-point method*, Nauka, Moscow, 1977. (Russian) MR**0507923 (58:22580)****11.**E. Pelinovsky and R. Mazova, Natural Hazards**6**(1992), 227-249.**12.**V. M. Babich, V. S. Buldyrev, and I. A. Molotkov,*The space-time ray method. Linear and nonlinear waves*, Leningrad. Univ., Leningrad, 1985. (Russian) MR**0886885 (88e:35002)****13.**V. M. Babich,*Propagation of non-stationary waves and caustics*, Leningrad. Gos. Univ. Uchen. Zap. Ser. Mat.**32**(1958), 228-260. (Russian) MR**0111295 (22:2159)****14.**S. Yu. Dobrokhotov and B. Tirozzi,*Localized solutions of a one-dimensional nonlinear system of shallow water equations with velocity*, Uspekhi Mat. Nauk**65**(2010), no. 1, 185-186; English transl., Russian Math. Surveys**65**(2010), no. 1, 177-179. MR**2655250****15.**M. V. Berry,*Focused tsunami waves*, Proc. Roy. Soc. London Ser. A**463**(2007), 3055-3071. MR**2360189 (2009b:86002)****16.**V. S. Vladimirov,*The equations of mathematical physics*, 4th. ed., Nauka, Moscow, 1981; English transl., Mir, Moscow, 1984. MR**0653331 (83i:00029)**; MR**0764399 (86f:00030)****17.**A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi,*Higher transcendental functions*. Vol. II, Robert E. Krieger Publ. Co., Inc., Melbourne, FL, 1981. MR**0698779 (84h:33001a)****18.**M. Sh. Birman and M. Z. Solomyak,*Spectral theory of selfadjoint operators in Hilbert space*, Leningrad. Univ., Leningrad, 1980; English transl., D. Reidel Publ. Co., Dordrecht, 1987. MR**0609148 (82k:47001)**; MR**1192782 (93g:47001)****19.**V. I. Arnold,*Mathematical methods of classical mechanics*, Nauka, Moscow, 1974; English transl., Grad. Texts in Math., vol. 60, Springer-Verlag, New York, 1989. MR**0474390 (57:14032)**; MR**0997295 (90c:58046)****20.**S. F. Dotsenko, B. Yu. Sergeevskiĭ, and L. B. Cherkesov,*Space tsunami waves induced by sign-variable dislocation of the ocean surface*, Investigation of Tsunami, No. 1, Mezhduved. Geofiz. Komitet pri Prezideume Akad. Nauk SSSR, Moscow, 1986, pp. 7-14. (Russian)**21.**S. Wang,*The propagation of the leading wave*, ASCE Specialty Conference on Coastal Hydrodynamics (Univ. Delaware, June 29 -- July 1), 1987, pp. 657-670.**22.**S. Ya. Sekerzh-Zenkovich,*Simple asymptotic solution to the Cauchy-Poisson problem for head waves*, Russ. J. Math. Phys.**16**(2009), no. 2, 315-322. MR**2525418 (2010e:35043)**

Retrieve articles in *St. Petersburg Mathematical Journal*
with MSC (2010):
35L05

Retrieve articles in all journals with MSC (2010): 35L05

Additional Information

**S. Yu. Dobrokhotov**

Affiliation:
Institute for Problems in Mechanics, Russian Academy of Sciences, and Moscow Institute of Physics and Technology, Russia

Email:
dobr@ipmnet.ru

**V. E. Nazaĭkinskiĭ**

Affiliation:
Institute for Problems in Mechanics, Russian Academy of Sciences, and Moscow Institute of Physics and Technology, Russia

Email:
nazay@ipmnet.ru

**B. Tirozzi**

Affiliation:
Department of Physics, University of Rome “La Sapienza”, Italy

Email:
brunello.tirozzi@roma1.infn.it

DOI:
https://doi.org/10.1090/S1061-0022-2011-01175-6

Keywords:
Wave equation with degenerating velocity,
asymptotic expansion,
wave front,
singular Lagrangian manifold,
run-up

Received by editor(s):
September 13, 2010

Published electronically:
August 18, 2011

Additional Notes:
Supported by RFBR (grants nos. 08-01-00726 and 09-01-12063-ofi-m) and by a joint project of the Department of Physics of the University of Rome “La Sapienza” and the Institute for Problems in Mechanics, Russian Academy of Sciences

Dedicated:
To Vasiliĭ Mikhaĭlovich Babich

Article copyright:
© Copyright 2011
American Mathematical Society