Abstract
In this chapter we study the existence of nontrivial water waves in a channel of finite depth. As shown in Figure 71.1 we find that, in addition to the trivial parallel flow, there occur nontrivial wave motions at certain critical velocities c. Such waves were studied during the nineteenth century by British hydro-dynamicists such as Airy, Stokes, Kelvin, and Rayleigh. They solved the linearized problems and calculated nonlinear approximations up to order 6. No convergence proofs, however, were given.
In one of his last papers, Lord Rayleigh, in 1917, computed approximate solutions up to order 6. He also showed by means of a numerical example that the relative error is not greater than 2.5 10-6. We give here a rigorous existence proof for permanent gravitational waves of infinite depth which is also constructive.
Tullio Levi-Civita (1925)
I have tried to avoid long numerical computations, thereby following Riemann’s postulate that proofs should be given through ideas and not voluminous computations.
David Hilbert, Report on Number Theory (1897)
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References to the Literature
Classical works: Nekrasov (1921), Levi-Civita (1925), Lavrentjev (1946),and Friedrichs and Hyers (1954) (solitary waves).
Monograph about permanent waves: Zeidler (1968, B, H).
Permanent waves: Zeidler (1977, S), (1971), (1972a, b), (1973), Beyer and Zeidler (1979), Beale (1979), Turner (1981), Amick and Toland (1981), Amick, Fraenkel, and Toland (1982)(proof of the Stokes conjecture).
The blowing-up lemma and applications of bifurcation theor.y to capillary-gravity waves: Zeidler (1968, M), Jones and Toland (1986).
Homoclinic bifurcation of dynamical systems and permanent waves: Kirchgassner (1988).
Survey about wave problems in physics: Whitham (1974, M, B) (standard work), Stoker (1957, M) (water waves), Lighthill (1978, M) (1986),leBlond (1978, M) (waves in the ocean), Friedlander (1981) (geophysics), Debnath (1985, P) (nonlinear waves), Brekhovskikh (1985, M), Ghil (1987, M) (atmospheric dynamics, dynamo theory, and climate dynamics), Washington (1987, M) (climate modelling).
Dead water problems and cavities: Leray (1935) (classical work), Serrin (1952), Birkkhoff and Zarantonello (1957, M), Hilbig (1964),(1982),Socolescu (1977).
Variational approach to jets and cavities: Friedman (1987, M) (recommended as an introduction), Alt and Caffarelli (1981).
Equilibrium forms of rotating fluids: Lichtenstein (1933, M) (classical monograph), Chandrasekhar (1969, M), Lebovitz (1977, S), Friedman (1982, M).
Rotating stars: Tassoul (1978, M).
Solitons: Novikov (1980, M) and Ablowitz and Segur (1981, M, H) (recommended as an introduction), Lax (1968), Bullough and Caudrey (1980, P) (applications in physics), Lamb (1980, M), Calogero and Degasperis (1982, M), Drazin (1983, M), Rajaraman (1983, M), (solitons, instantons, and elementary particles),Davydov (1984, M) (solitons in molecular systems), Faddeev and Takhtadjan (1986, M) (infinitedimensional Hamiltonian systems, solitons, the Riemann-Hilbert problem, and inverse scattering theory), Knorrer (1986, S) (solitons and algebraic geometry).
Collection of classical articles on solitons: Rebbi (1984).
The initial-value problem for the Korteweg-de Vries equation: Bona and Smith (1975), Kato (1983).
The initial-value problem for water waves: Shinbrot (1976).
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© 1988 Springer Science+Business Media New York
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Zeidler, E. (1988). Bifurcation and Permanent Gravitational Waves. In: Nonlinear Functional Analysis and its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4566-7_15
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DOI: https://doi.org/10.1007/978-1-4612-4566-7_15
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