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The Korteweg-de Vries equation: a historical essay

Published online by Cambridge University Press:  20 April 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla, California 92093
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Abstract

The Korteweg-de Vries (KdV) equation, usually attributed to Korteweg & de Vries (1895), governs the propagation of weakly dispersive, weakly nonlinear water waves and serves as a model equation for any physical system for which the dispersion relation for frequency vs. wavenumber is approximated by ω/k = c0(1 − βk2) and nonlinearity is weak and quadratic. It first appears explicitly in de Vries's dissertation (1894), although it is implicit in the work of Boussinesq (1872). Its current renaissance stems from the Fermi, Pasta & Ulam (1955) problem for a string of nonlinearly coupled oscillators, which, through the work of Zabusky & Kruskal and their colleagues, led to the discovery of the soliton and the development of inverse-scattering theory by Gardner et al. (1967). Many related evolution equations, each of which represents a balance between some form of dispersion (or variation of dispersion in the case of wave-packet evolution) and weak nonlinearity in an appropriate reference frame, have since been found to have properties analogous to those of the KdV equation - in particular, inverse-scattering solutions that are asymptotically dominated by solitons.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 106 , May 1981 , pp. 131 - 147
Copyright
© 1981 Cambridge University Press

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References

Abe, K. & Abe, T. 1979 Recurrence of initial state of the Korteweg-de Vries equation. Phys. Fluids 22, 1644–1646.Google Scholar
Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. 1974 The inverse scattering transform—Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315.Google Scholar
Ablowitz, M. J. & Kruskal, M. 1979 A note on Miura's transformation. J. Math. Phys. 20, 999–1003.Google Scholar
Abramowitz, M. & Stegun, I. 1964 Handbook of Mathematical Functions. Washington: National Bureau of Standards.
Bazin, H. 1865 Recherches expérimentales sur la propagation des ondes. Mém. présentés par divers Savants á L'Acad. Sci. Inst. France 19, 495–644.Google Scholar
Benjamin, T. B. 1967 Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29, 559–592.Google Scholar
Benjamin, T. B., Bona, J. L. & Mahony, J. J. 1972 Model equations for long waves in non-linear dispersive systems. Phil. Trans. Roy. Soc. A 272, 47–78.Google Scholar
Benjamin, T. B. & Mahony, J. J. 1971 On an invariant property of water waves. J. Fluid Mech. 49, 385–389.Google Scholar
Beth, H. & Van der Woude, W. 1946 Levensbericht van D. J. Korteweg (31 Maart 1848-10 Mai 1941). Jaarboek Kon. Ned. Akad. v. Wetenschappen, 1945–46. Amsterdam.
Boussinesq, J. 1871a Théorie de l'intumescence liquide, appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. C. r. Acad. Sci. Paris 72, 755–759.Google Scholar
Boussinesq, J. 1871b Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal. C. r. Acad. Sci. Paris 73, 256–260.Google Scholar
Boussinesq, J. 1872 Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilies de la surface au fond. J. Math. Pures Appl. 17 (2), 55–108; see also Vastano & Mungall (1976).Google Scholar
Boussinesq, J. 1877 Essai sur la théorie des eaux courantes. Mém. présentés par divers Savants à L'Acad. Sci. Inst. France (séries 23, 1–680 (see also 24, 1–64).Google Scholar
Broer, L. J. F. 1974 On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29, 430–446.Google Scholar
Broer, L. J. F. 1975a Hidden Hamiltonians of first-order equations. Physica A 79, 583–596.Google Scholar
Broer, L. J. F. 1975b Approximate equations for long water waves. Appl. Sci. Res. 31, 377–395.Google Scholar
Broer, L. J. F. 1976 Stable model equations for long water waves. Appl. Sci. Res. 32, 619–636.Google Scholar
Chen, H. H., Hirota, R. & Lee, Y. C. 1980 Inverse scattering problem for internal waves with finite fluid depth. Phys. Lett. A 75, 254–256.Google Scholar
de Vries, G. 1894 Bijdrage tot de kennis der lange golven. Doctoral dissertation, University of Amsterdam.
Fermi, E., Pasta, J. & Ulam, S. 1955/1974 Studies of nonlinear problems. I. Nonlinear Wave Motion. Lectures in Applied Mathematics, vol. 15 (ed. A. C. Newell), pp. 143–156. Amer. Math. Soc., Providence, R.I.
Flaschka, H. & Newell, A. 1975 Integrable systems of nonlinear evolution equations. Dynamical Systems, Theory and Applications (ed. J. Moser), pp. 355–440. Springer.
Fornberg, B. & Whitham, G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena. Phil. Trans. Roy. Soc. A 289, 373–404.Google Scholar
Friedrichs, K. O. & Hyers, D. H. 1954 The existence of solitary waves. Comm. Pure Appl. Math. 7, 517–550.Google Scholar
Gardner, C. S. 1971 Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys. 12, 1548–1551.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–97.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1974 Korteweg-de Vries equation and generalizations. VI. Methods for exact solution. Comm. Pure Appl. Math. 27, 97–133.Google Scholar
Gillispie, C. C. 1970 Dictionary of Scientific Biography. Scribners.
Hammack, J. L. & Segur, H. 1974 The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments. J. Fluid Mech. 65, 289–314.Google Scholar
Henyey, F. S. 1980 Finite-depth and infinite-depth internal-wave solitons. Phys. Rev. A 21, 1054–1056.Google Scholar
Hirota, R. 1971 Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194.Google Scholar
Hirota, R. 1976 Direct method of finding exact solutions of nonlinear evolution equations. Bácklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications (ed. R. M. Miura), pp. 40–68. Springer.
Jeffrey, A. & Kakutani, T. 1972 Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation. SIAM Rev. 14, 582–643.Google Scholar
Joseph, R. I. 1977 Solitary waves in a finite-depth fluid. J. Phys. A 10, 225–227.Google Scholar
Karpman, V. I. 1979 Soliton evolution in the presence of perturbation. Physica Scripta 20, 462–48.Google Scholar
Karpman, V. I. & Maslov, E. M. 1977 A perturbation theory for the Korteweg-de Vries equation. Phys. Lett. A 60, 307–308.Google Scholar
Kaup, D. J. & Newell, A. C. 1978 Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory. Proc. Roy. Soc. A 361, 413–446.Google Scholar
Korteweg, D. J. & De Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39, 422–443.Google Scholar
Kruskal, M. D. 1974 The Korteweg-de Vries equation and related evolution equations. Nonlinear Wave Motion. Lectures in Applied Mathematics, vol. 15 (ed. A. C. Newell), pp. 61–84. Am. Math. Soc., Providence, R.I.
Kruskal, M. D. 1975 Nonlinear wave equations. Dynamical Systems Theory and Applications (ed. J. Moser), pp. 310–354. Springer.
Kruskal, M. D. 1978 The birth of the soliton. Nonlinear Evolution Equations Solvable by the Spectral Transform (ed. Francesco Calogero), pp. 1–8. Pitman.
Laitone, E. V. 1960 The second approximation to cnoidal and solitary waves. J. Fluid Mech. 9, 430–444.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lax, P. D. 1968 Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math. 21, 467–490.Google Scholar
Lax, P. D. 1976 Almost periodic solutions of the KdV equation. SIAM Rev. 18, 351–375.Google Scholar
Makhankov, V. G. 1978 Dynamics of classical solitons (in non-integrable systems). Phys. Rep. 35, 1–128.Google Scholar
Miles, J. W. 1976 Korteweg-de Vries equation modified by viscosity. Phys. Fluids 19, 1063.Google Scholar
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Miles, J. W. 1979 On internal solitary waves. Tellus 31, 456–462.Google Scholar
Miles, J. W. 1980 Solitary waves. Ann. Rev. Fluid Mech. 12, 11–43.Google Scholar
Miura, R. M. 1968 Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys. 9, 1202–1204.Google Scholar
Miura, R. M. 1974 The Korteweg-de Vries equation: a model equation for nonlinear dispersive waves. Nonlinear Waves (eds. S. Leibovich & A. R. Seebass). pp. 212–234. Cornell University Press.
Miura, R. M. 1976 The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18, 412–459.Google Scholar
Miura, R. M., Gardner, C. S. & Kruskal, M. D. 1968 Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion. J. Math. Phys 9, 1204–1209.Google Scholar
Ono, H. 1975 Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39, 1082–1091.Google Scholar
Ostrovsky, L. A. & Pelinovsky, E. N. 1975 Refraction of nonlinear ocean waves in a beach zone. Atmos. & Oceanic Phys. 11, 37–41.Google Scholar
Peregrine, D. H. 1966 Calculations of the development of an undular bore. J. Fluid Mech. 25, 321–330.Google Scholar
Rayleigh, Lord 1876 On waves. Phil. Mag. 1, 257–279; Scientific Papers, vol. 1, pp. 251–271.Google Scholar
Rouse, H. & Ince, S. 1957 History of Hydraulics. Dover.
Russell, J. S. 1845 Report on waves. Rep. Meet. Brit. Assoc. Adv. Sci., 14th, York, 1844, pp. 311–390. London: John Murray.
Satsuma, J., Ablowitz, M. & Kodama, Y. 1979 On an internal wave equation describing a stratified fluid with finite depth. Phys. Lett. A 73, 283–286.Google Scholar
Scott, A. C., Chu, F. Y. F. & McLaughlin, D. W. 1973 The soliton: a new concept in applied science. Proc. I.E.E.E. 61, 1443–1483.Google Scholar
Shuto, N. 1974 Non-linear long waves in a channel of variable section. Coastal Eng. Japan 17, 1–12.Google Scholar
Stoker, J. J. 1957 Water Waves. Interscience.
Tuck, J. L. & Menzel, M. T. 1972 The superperiod of the nonlinear weighted string (FPU) problem. Advances in Maths 9, 399–407.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685–694.Google Scholar
van der Blij, F. 1978 Some details of the history of the Korteweg-de Vries equation. Nieuw Archief Voor Wiskunde 26, 54–64.Google Scholar
Vastano, A. C. J. & Mungall, J. C. H. 1976 Theory of waves and surges which propagate the length of a horizontal rectangular canal, imparting to the fluid contained within the canal velocities that are sensibly the same from the top to the bottom [English translation of Boussinesq (1872)]. College of Geosciences, Texas A & M Univ. Rep. 76-2-T.
Wadati, M. 1972 The exact solution of the modified Korteweg-de Vries equation. J. Phys. Soc. Japan 32, 1681.Google Scholar
Wadati, M. 1973 The modified Korteweg-de Vries equation. J. Phys. Soc. Japan 34, 1289–1296.Google Scholar
Wahlquist, H. D. & Estabrook, F. B. 1973 Bäcklund transformations for solutions of the Korteweg-de Vries equation. Phys. Rev. Lett. 31, 1386–1390.Google Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface waves. Encyc. Phys. 9, 446–778.Google Scholar
Whitham, G. B. 1965 Non-linear dispersive waves. Proc. Roy. Soc. A 283, 238–261.Google Scholar
Whitham, G. B. 1967 Variational methods and applications to water waves. Proc. Roy. Soc. A 299, 6–25.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Zabusky, N. J. & Galvin, C. J. 1971 Shallow-water waves, the Korteweg-de Vries equation and solitons. J. Fluid Mech. 47, 811–824.Google Scholar
Zabusky, N. J. & Kruskal, M. D. 1965 Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243.Google Scholar
Zabusky, N. J., Kruskal, M. D. & Deem, G. S. 1965 Formation, Propagation and Interaction of Solitons (Numerical Solutions of Differential Equations Describing Wave Motion in Nonlinear Dispersive Media). A 16 mm, 35 min silent, black-and-white film available on loan from the Bell Telephone Laboratories, Inc. Film Library, Murray Hill, N.J. 07971.
Zakharov, V. E. 1974 On stochastization of one-dimensional chains of nonlinear oscillators. Sov. Phys., J. Exp. Theor. Phys. 38, 108–110.Google Scholar
Zakharov, V. E. & Faddeev, L. D. 1972 Korteweg-de Vries equation: a completely integrable Hamiltonian system. Functional Anal. Appl. 5, 280–287.Google Scholar
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys., J. Exp. Theor. Phys 34, 62–69.Google Scholar