Abstract
This survey presents new mathematical results in the theory of linear and nonlinear waves on the surface of a flotation liquid. A flotation liquid is a liquid on whose surface heavy particles are floating; the particles may consist of arbitrary materials or may be particles of frozen liquid.
The first part of the article considers initial- and boundary-value problems in the theory, their solvability, and the behavior of the solutions over long periods. In the second part of the survey, theorems are proved on the existence of nonlinear standing waves within the framework of an exact physical model, and both internal and free waves are considered. Also, the fundamental equations for shallow flotation waves are derived and examined.
Similar content being viewed by others
Literature cited
A. Erdelyi (ed.), Higher Transcendental Functions, McGraw-Hill, New York (1953).
Yu. M. Berezanskii, “Expansion of Self-adjoint Operators in Characteristic Functions” [in Russian], Naukova Dumka, Kiev (1965).
M. M. Vainberg and V. A. Trenogin, The Theory of Branching of Solutions to Nonlinear Equations [in Russian], Nauka, Moscow (1969).
V. S. Vladimirov, The Equations of Mathematical Physics [in Russian], Nauka, Moscow (1971).
S. A. Gabov, “One problem on the hydrodynamics of an ideal liquid associated with flotation,” Differents. Uravnen.,24, No. 1, 16–21 (1988).
S. A. Gabov, “The existence of standing waves of finite amplitude on the surface of a flotation liquid,” Zh. Vychisl. Mat. Mat. Fiz.,28, No. 10, 1507–1519 (1988).
S. A. Gabov and A. G. Sveshnikov, Problems in the Dynamics of Stratified Liquids [in Russian], Nauka, Moscow (1986).
S. A. Gabov and A. G. Sveshnikov, “Mathematical problems in the dynamics of stratified liquids,” in: Mathematical Modelling: Processes in Nonlinear Media [in Russian], Moscow (1986), pp. 107–141.
F. D. Gakhov, Boundary-Value Problems [in Russian], Fizmatgiz, Moscow (1963).
I. M. Gel'fand and G. E. Shilov, Generalized Functions and Operations on Them [in Russian], Gos. Izd. Fiz.-Mat. Lit., Moscow (1958).
N. M. Gunter, Potential Theory and Its Application to Basic Problems of Mathematical Physics [in Russian], Gostekhizdat, Moscow (1953).
S. Yu. Dobrokhotov and P. N. Zhevandrov, “Nonstandard characteristics and Maslov's operator method in linear problems on nonstanding water waves,” Funkts. Anal. Ego. Prilozehn.,19, No. 4, 43–54 (1985).
P. N. Zhevandrov, “Wakes on the surface of a flotation liquid,” Zh. Vychisl. Mat. Mat. Fiz.,28, No. 6, 1110–1115 (1988).
P. P. Zabreiko, A. I. Koshelev, M. A. Krasnosel'skii, S. G. Mikhlin, L. S. Rakovshchik, and V. Ya. Stetsenko, Integral Equations [in Russian], Nauka, Moscow (1968).
F. Calogero and A. Degasperis, Spectral Transformations and Solitons, Elsevier, New York (1982).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1976).
E. T. Copson, Asymptotic Expansions, Cambridge Univ. Press, Cambridge (1965).
M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations [in Russian], Gostekhizdat, Moscow (1956).
M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate Solution of Operator Equations [in Russian], Nauka, Moscow (1969).
S. G. Krein, Linear Differential Equations in Banach Space [in Russian], Nauka, Moscow (1967).
S. G. Krein (ed.), Functional Analysis [in Russian], Nauka, Moscow (1972).
O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).
H. Lamb, Hydrodynamics, Cambridge Univ. Press, Cambridge (1932).
J. L. Lions and E. Madjenes, Nonhomogeneous Boundary-Value Problems and their Application, Springer-Verlag, New York (1972).
S. G. Mikhlin, Linear Partial Differential Equations [in Russian], Vysshaya Shkola, Moscow (1977).
S. G. Mikhlin, Multidimensional Integrals and Integral Equations [in Russian], Fizmatgiz, Moscow (1962).
N. N. Moiseev and V. V. Rumyantsev, The Dynamics of Solids With Liquid Filled Cavities [in Russian], Nauka, Moscow (1965).
V. N. Monakhov, Boundary-Value Problems with Free Broundaries for Elliptic Systems [in Russian], Nauka, Novosibirsk (1977).
N. I. Muskhelishvili, Singular Integral Equations. Boundary-Value Problems in the Theory of Functions and Some of Their Applications to Mathematical Physics [in Russian], Fizmatgiz, Moscow (1962).
A. I. Nekrasov, The Exact Theory of Standing Waves on the Surface of a Heavy Liquid. Collected Works. Vol. 1 [in Russian], Fizmatgiz, Moscow (1961).
B. Noble, Methods Based on The Wiener-Hopf Technique for the Solution of Partial Differential Equations, Pergamon, New York (1959).
L. V. Ovsyannikov, N. I. Makarenko, V. I. Nalimov, et. al., Nonlinear Problems in the Theory of Surface and Internal Waves [in Russian], Nauka, Novosibirsk (1985).
F. Riesz and B. Sz. Nagy, Functional Analysis [Russian translation], Izd. In. Lit., Moscow (1954).
L. N. Sretenskii, The Theory of Wave Motions of Liquids [in Russian], Nauka, Moscow (1977).
J. J. Stoker, Water Waves, Wiley, New York (1957).
A. N. Tikhonov and A. A. Samarskii, The Equations of Mathematical Physics [in Russian], Nauka, Moscow (1966).
G. Whitham, Linear and Nonlinear Waves, Wiley, New York (1974).
M. V. Fedoryuk, Asymptotics. Integrals and Series [in Russian], Nauka, Moscow (1987).
P. Halmos, A Hilbert Space Problem Book, Van Nostrand-Reinhold, New York (1967).
R. M. Garipov, “On the linear theory of gravity waves: the theorem of existence and uniqueness,” Arch. Ration. Mech. Anal.,24, No. 5, 352–362 (1967).
B. N. Mandai, “Water waves generated by disturbances at an inertial surface,” Appl. Sci. Res.,45, No. 1, 67–73 (1988).
B. N. Mandai and Kundu Krishna, “A note on the singularities in the theory of water waves with an inertial surface,” J. Aust. Math. Soc.,28, No. 2, 271–278 (1986).
A. S. Petters, “The effect of a floating mat on water waves,” Commun. Pure Appl. Math.,3, No. 4, 319–354 (1950).
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 28, pp. 3–86, 1990.
Rights and permissions
About this article
Cite this article
Gabov, S.A., Sveshnikov, A.G. Problems in the dynamics of flotation liquids. J Math Sci 54, 979–1041 (1991). https://doi.org/10.1007/BF01138947
Issue Date:
DOI: https://doi.org/10.1007/BF01138947