Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T13:40:26.456Z Has data issue: false hasContentIssue false
  • English
  • Français

The stability of internal solitary waves

Published online by Cambridge University Press:  24 October 2008

D. P. Bennett ,
R. W. Brown ,
S. E. Stansfield ,
J. D. Stroughair  and
J. L. Bona
D. P. Bennett
Affiliation:
Physics Department, Case Western Reserve University, Cleveland, Ohio
R. W. Brown
Affiliation:
Physics Department, Case Western Reserve University, Cleveland, Ohio
S. E. Stansfield
Affiliation:
Physics Department, Case Western Reserve University, Cleveland, Ohio
J. D. Stroughair
Affiliation:
Physics Department, Case Western Reserve University, Cleveland, Ohio
J. L. Bona
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois
Get access Rights & Permissions [Opens in a new window]

Extract

A theory is developed relating to the stability of solitary-wave solutions of the so-called Benjamin-Ono equation. This equation was derived by Benjamin (5) as a model for the propagation of internal waves in an incompressible non-diffusive heterogeneous fluid for which the density is non-constant only within a layer whose thickness is much smaller than the total depth. In his article, Benjamin wrote in closed form the one-parameter family of solitary-wave solutions of his model equation whose stability will be the focus of attention presently.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 2 , September 1983 , pp. 351 - 379
Copyright
Copyright © Cambridge Philosophical Society 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Ablowitz, M. J., KAUP, D. J., Newell, A. C. and Segur, H.Nonlinear-evolution equations of physical significance Phys. Rev. Lett. 31 (1973), 125.CrossRefGoogle Scholar
(2)Adams, R. A.Sobolev spaces. (Academic Press, 1975).Google Scholar
(3)Apel, J. R.Observations of internal-wave surface signatures in ASTP photographs. Apollo Soyuz Test Project, Summary Sci. Rep. NASA SP-412 (1979).Google Scholar
(4)Apel, J. R., Byrne, H. M., PRONI, J. E. and Charnell, R. L.Observations of oceanic and internal surface waves from the Earth Resources Technology Satellite. J. Geophys. Res. 80 (1975), 865.Google Scholar
(5)Benjamin, T. B.Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29 (1967), 559.CrossRefGoogle Scholar
(6)Benjamin, T. B.The stability of solitary waves. Proc. R. Soc. Lond. Ser. A 328 (1972), 153.Google Scholar
(7)Benjamin, T. B., BONA, J. L. and BOSE, D. K.Solitary-wave solutions for some model equations for waves in nonlinear dispersive media. In the proceedings of the joint IUTAM/IMU symposium on applications of methods of functional analysis to problems in mechanics, Marseille, Sept. 1975. Springer Lecture Notes in Mathematics No. 503 (1976), 207.Google Scholar
(8)Benjamin, T. B., Bona, J. L. and Bose, D. K. Fixed-point theorems in Frchet spaces and the existence of solitary waves. (Preprint.)Google Scholar
(9)Benjamin, T. B., Bona, J. L. and Mahonv, J. J.Model equations for long waves in non-linear dispersive systems. Phil. Trans. R. Soc. Lond. Ser. A 272 (1972), 47.Google Scholar
(10)Bona, J.On the stability theory of solitary waves. Proc. R. Soc. Lond. Ser. A 344 (1975), 363.Google Scholar
(11)Bona, J. L. and Felland, M. E.The initial-value problem for wave equations having general forms of non-linearity and dispersion. In preparation.Google Scholar
(12)Bona, J. L. and Bose, D. K.Solitary-wave solutions for unidirectional wave equations having general forms of nonlinearity and dispersion. Fluid Mechanics Research Institute Report no 99 (1978).Google Scholar
(13)Bona, J. L., BOSE, D. K. and Turner, R. E. L.Finite-amplitude steady waves in stratified fluids. J. Math. Pures Appl. (in the Press).Google Scholar
(14)Bona, J. L., Prttchabd, W. G. and SCOTT, L. R.Solitary wave interaction. Phys. Fluids 23 (1980), 438.CrossRefGoogle Scholar
(15)Bona, J. L. and Smith, R.The initial-value problem for the Korteweg-de Vries equation. Phil. Trans. R. Soc. Lond. Ser. A 278 (1975), 555.Google Scholar
(16)Chen, H. H. & Kaup, D. J.Linear stability of internal wave solitons in a deep stratified fluid. Phys. Fluids 23 (1980), 235.Google Scholar
(17)Davis, R. E. and ACRIVOS, A.Solitary internal waves in deep water. J. Fluid Mech. 29 (1967), 593.Google Scholar
(18)Djordjevic, V. D. and Redekopp, L. G.The fission and disintegration of internal solitary waves moving over two-dimensional topography. J. Phys. Oceanogr. 8 (1978), 1016.2.0.CO;2>CrossRefGoogle Scholar
(19)Dunford, N. and Schwartz, J. T.Linear operators part II: spectral theory, self adjoint operators in Hilbert space (Interscience, 1963).Google Scholar
(20)Joseph, R. I.Solitary waves in a finite depth fluid. J. Phys. A 10 (1977), L 225.Google Scholar
(21)Kato, T.Perturbation theory for linear operations, 2nd ed. (Springer-Verlag, 1976).Google Scholar
(22)Lions, J. L. and Magenes, E.Problmes aux limites non homognes et applications, vol. 1 (Dunod, 1968).Google Scholar
(23)Maxworthy, T.A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge. J. Geophys. Res. 84 (1979), 338.CrossRefGoogle Scholar
(24)Maxworthy, T.On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dimensions. J. Fluid Mech. 96 (1980), 47.Google Scholar
(25)Mckean, H. P.Stability for the Korteweg-de Vries equation. Comm. Pure Appl. Math. 30 (1977), 347.CrossRefGoogle Scholar
(26)Mcleod, J. B. and OLVER, P. J.The connection between partial differential equations solvable by inverse scattering and ordinary differential equations of Painleve type. Siam J. Math. Anal. 14 (1983), 488.CrossRefGoogle Scholar
(27)Olver, P. J.Euler operators and conservation laws of the BBM equation. Math. Proc. Cambridge Phil. Soc. 85 (1979), 143.CrossRefGoogle Scholar
(28)Osborne, A. R. and BURCH, T. L.Internal solitons in the Andaman sea. Science 208 (1980), 451.CrossRefGoogle ScholarPubMed
(29)Peregrine, D. H.Calculations of the development of an undular bore. J. Fluid Mech. 25 (1964), 321.CrossRefGoogle Scholar
(30)Satsuma, J., Ablowitz, M. J. and Kodama, Y.On an internal wave equation describing a stratified fluid with finite depth. Phys. Lett. 73A (1979), 283.CrossRefGoogle Scholar
(31)Scott, A. C, CHU, F. Y. F. and Mclaughlin, D. W.The soliton: a new concept in applied science. Proc. Ieee 61 (1973), 1443.Google Scholar
(32)Stein, E. M.Singular integrals and differentiability properties of functions. Princeton Mathematical Series, no. 30 (Princeton University Press, 1970).Google Scholar
(33)Ter-Krikerov, A. M.Theorie exacte des ondes longues stationnaires dans un liquide htrogne. J. Mcanique 2 (1963), 351.Google Scholar
(34)Titchmarsh, E. C.Introduction to the theory of Fourier integrals (Oxford University Press, 1937).Google Scholar
(35)Titchmarsh, E. C.Eigenfunction expansions, 2nd ed. (Oxford University Press, 1962).Google Scholar
(36)Treves, F.Topological vector spaces, distributions and kernels (Acacemic Press, 1967).Google Scholar
(37)Turner, R. E. L.Internal waves in fluids with rapidly varying density. Ann. Scuola Norm. Sup. Pisa Ser. 4, 8 (1981), 513.Google Scholar
(38)Walker, L. R.Interfacial solitary waves in a two-fluid medium. Phys. Fluids 16 (1973), 1796.CrossRefGoogle Scholar
(39)Zabreyko, Z. Z., Koshelev, A. A., Krasnosel'Skii, M. A., Mikhlin, S. G., Rakovschik, L. S. and Stet'Senko, V. Ya.Integral equations a reference text (Noordhoff, 1975).Google Scholar