Abstract
We construct exact solutions describing trapped water waves over an underwater ridge of small height in the shallow water approximation and in the complete formulation. Resonances (antibound states) in the case of an underwater trench are also constructed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publ, Inc., New York, 1970).
S. Yu. Dobrokhotov, “Maslov’s Methods in the Linearized Theory of Gravitational Waves on the Fluid Surface,” Dokl. Akad. Nauk SSSR 269(1), 76–80 (1983) [Sov. Phys. Dokl., 28, 189–191 (1983).
S. Yu. Dobrokhotov, “Asymptotic Behavior of Water Surface Waves Trapped by Shores and Irregularities of the Bottom Relief,” Dokl. Akad. Nauk SSSR 289(3), 575–579 (1986) [Soviet Phys. Dokl. 31 (7), 537–539 (1986)].
R. Gadyl’shin, “Local Perturbations of the Schrödinger Operator on the Axis,” Theor. Math. Phys. 132, 976–982 (2002).
D. S. Kuznetsov, “A Spectral Perturbation Problem and Its Applications to Waves above an Underwater ridge,” Siberian Math. J., 42, 668–684 (2001).
L. D. Landau, E. M. Lifshitz, “Quantum Mechanics. Non-Relativistic Theory,” Pergamon (1958).
M. A. Lavrentiev, and B. V. Chabat, “Effets Hydrodynamiques et Modèles Mathématiques,” (Mir, Moscow, 1972).
D. B. Nicholls and M. Taber, “Joint Analyticity and Analytic Continuation of Dirichlet-Neumann Operators on Doubly Perturbed Domains,” J. Math. Fluid Mech., 10, 238–271 (2008).
L. V. Ovsjannikov [Ovsyannikov], “Cauchy Problem in a Scale of Banach Spaces and Its Application to the Shallow Water Theory Justification,” in: Applications of Methods of Functional Analysis to Problems in Mechanics, P. Germain and B. Nayroles (Editors), Lect. Notes Math., 503 (Springer, Berlin-New York, 1976), pp. 426–437.
L. V. Ovsyannikov, N. I. Makarenko, V. I. Nalimov, V, Yu, Lyapidevskii, P. I. Plotnikov, I. V. Sturova, V. I. Bukreev, and V. A. Vladimirov, “Nonlinear Problems in the Theory of Surface and Internal Waves” (Nauka Sibirsk. Otdel., Novosibirsk, 1985) (in Russian).
J. A. Rodriguez Ceballos and P. Zhevandrov, “Comparison between the Discrete and Continuous Schrödinger Operators with a Small Potential Well,” in Numerical Modelling of Coupled Phenomena in Engineering and Science: Practical Uses and Examples, M.C. Suárez Arriaga, J. Bundschuh, F.J. Domínguez-Mota (Editors) (Taylor & Francis, 2008), pp. 71–88.
B. Simon, “The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions,” Ann. Phys., 97, 279–288 (1976).
B. Simon, “Resonances in One Dimension and Fredholm Determinants,” J. Funct. Anal., 178, 396–420 (2000).
I. Stakgold, “Boundary Value Problems of Mathematical Physics” (Macmillan Company, Vol. II, New York, 1968).
E. C. Titchmarsh, The Theory of Functions (Oxford University Press, 1939).
B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics (Moskov. Gos. Univ., Moscow, 1982; Gordon & Breach Science Publishers, New York, 1989).
V. S. Vladimirov, Equations of Mathematical Physics (Mir Publishers, Moscow, 1984).
P. Zhevandrov and A. Merzon, “Asymptotics of Eigenfunctions in Shallow Potential Wells and Related Problems,” Amer. Math. Soc. Trans. (2) 208, 235–284 (2003).
M. Zworski, “Resonances in Physics and Geometry,” Notices Amer. Math. Soc. 46(3), 319–328 (1999).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Romero Rodríguez, M.I., Zhevandrov, P. Trapped modes and resonances for water waves over a slightly perturbed bottom. Russ. J. Math. Phys. 17, 307–327 (2010). https://doi.org/10.1134/S1061920810030052
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920810030052