Solitary waves and fundamental solution for Ostrovsky equation. (English) Zbl 1073.76011

Varlamov, Vladimir; Liu, Yue

Math. Comput. Simul. 69, No. 5-6, 567-579 (2005).

Summary: The Ostrovsky [L. A. Ostrovsky, Okeanologia 12, No. 2, 181–191 (1978)] equation describes the propagation of one-dimensional long waves in shallow water in the presence of rotation (Coriolis effect). In this model dispersion is taken into account and dissipation is neglected. It is proved that existence and non-existence of solitary waves depends on the sign of the dispersion parameter which can be either positive or negative. A fundamental solution of the linear Cauchy problem for Ostrovsky equation is constructed. Special function representation for it is obtained. Some properties of the fundamental solution are established, and its higher-order asymptotics is obtained as the rotation parameter tends to zero.

MSC:

76B25	Solitary waves for incompressible inviscid fluids
76U05	General theory of rotating fluids
35Q51	Soliton equations
86A05	Hydrology, hydrography, oceanography

Keywords:

shallow water; Coriolis effect; existence; dispersion parameter

References:

[1]	M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards, 1964.
[2]	Benilov, E.S., On the surface waves in a shallow channel with an uneven bottom, Stud. appl. math., 87, 1-14, (1992) · Zbl 0749.76007
[3]	Fróhlich, J.; Lieb, E.H.; Loss, M., Stability of Coulomb systems with magnetic fields. I. the one-electron atom, Commun. math. phys., 104, 251-270, (1986) · Zbl 0595.35098
[4]	Galkin, V.N.; Stepanyants, Yu.A., On the existence of stationary solitary waves in a rotating fluid, J. appl. math. mech., 6, 55, 939-943, (1991) · Zbl 0786.76016
[5]	Gilman, O.A.; Grimshaw, R.; Stepanyants, Yu.A., Approximate and numerical solutions of the stationary Ostrovsky equation, Stud. appl. math., 95, 115-126, (1995) · Zbl 0843.76008
[6]	Grimshaw, R., Evolution equations for weakly nonlinear long internal waves in a rotating fluid, Stud. appl. math., 73, 1-33, (1985) · Zbl 0572.76102
[7]	Kadomtsev, B.B.; Petviashvili, V.I., On the stability of solitary waves in weakly dispersive media, Sov. phys. dokl., 15, 6, 541-593, (1970) · Zbl 0217.25004
[8]	Lieb, E.H., On the lowest eigenvalue of Laplacian for the intersection of two domains, Invent. math., 74, 441-448, (1983) · Zbl 0538.35058
[9]	Lions, P.L., The concentration compactness principle in the calculus of variations. the locally compact case. parts 1 and 2, Ann. inst. H. Poincaré, anal. nonlinéaire, 1, (1984), 109-145, 223-283 · Zbl 0541.49009
[10]	Ostrovsky, L.A., Nonlinear internal waves in a rotating Ocean, Okeanologia, 18, 2, 181-191, (1978)
[11]	L.A. Ostrovsky, Yu.A. Stepanyants, Nonlinear surface and internal waves in rotating fluids, in: A.V. Gaponov-Grekhov, M.I. Rabinovich, J. Engelbrecht (Eds.), Research Reports in Physics, vol. 3, Nonlinear Waves, Springer, Berlin, Heidelberg, 1990. · Zbl 0719.76021
[12]	A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, vol. 1, Gordon and Breach, NY, 1990. · Zbl 0967.00503
[13]	V.K. Tuan, Airy integral transform and the Paley-Wiener theorem, in: Transform Methods & Special Functions, Proceedings of II International Workshop, Varna, 23-30 August 1996, pp. 523-531.
[14]	Varlamov, V.; Liu, Y., Cauchy problem for the Ostrovsky equation, Discrete cont. dynam. syst., 10, 731-753, (2004) · Zbl 1059.35035
[15]	Watson, G.N., A treatise on the theory of Bessel functions, (1980), Cambridge University Press NY · Zbl 0849.33001