Hamiltonian long-wave approximations to the water-wave problem. (English) Zbl 0929.76015
Summary: This paper presents a Hamiltonian formulation of the water-wave problem in which the non-local Dirichlet-Neumann operator appears explicitly in the Hamiltonian. The principal long-wave approximations for water waves are derived by the systematic approximation of the Dirichlet-Neumann operator by a sequence of differential operators obtained from a convergent Taylor expansion of the Dirichlet-Neumann operator. A simple and satisfactory method of obtaining the classical two-dimensional approximations such as the shallow-water, Boussinesq and KdV equations emerges from the process. A straightforward transformation theory describes the relationship between the classical symplectic structure appearing in the water-wave problem and the various non-classical symplectic structures that arise in long-wave approximations. The discussion extends to include three-dimensional approximations, including the KP equation.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 70H99 | Hamiltonian and Lagrangian mechanics |
References:
| [1] | Airy, G. B., Tides and waves, Encyc. Metropolitana, 5, 241-396 (1845) |
| [2] | Stokes, G. G., On the theory of oscillatory waves, Camb. Trans., 8, 441-473 (1847) |
| [3] | Rayleigh, Lord, On waves, Philos. Mag., 5, 1, 257-279 (1876) · JFM 08.0613.03 |
| [4] | Boussinesq, M. J., Théorie de l’intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire, Acad. Sci. Paris, Comptes Rendus, 72, 256-260 (1871) · JFM 03.0486.01 |
| [5] | Boussinesq, M. J., Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal, Acad. Sci. Paris, Comptes Rendus, 72, 256-260 (1871) · JFM 03.0486.02 |
| [6] | Boussinesq, M. J., 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 v itesses sensiblement pareilles de la surface au fond, J. Math. Pures Appl., 17, 55-108 (1872) · JFM 04.0493.04 |
| [7] | Boussinesq, M. J., Essai sur la théorie des eaux courantes, Mém présentés par divers Savants à L’Acad. Sci. Inst. France, 23, 1-680 (1872), (séries 2) · JFM 04.0493.03 |
| [8] | Korteweg, D. J.; de Vries, G., On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Philos. Mag., 5, 39, 422-443 (1895) · JFM 26.0881.02 |
| [9] | Benjamin, T. B., Impulse, flow-force and variational principles, IMA J. Appl. Math., 32, 3-68 (1984) · Zbl 0584.76001 |
| [10] | Russell, J. S., Report on waves, (Report of the Fourteenth Meeting of the British Association for the Advancement of Science (1844), John Murray: John Murray London), 311-390 |
| [11] | Zabusky, N. J.; Kruskal, M. D., Interactions of ‘solitons’ in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240-243 (1965) · Zbl 1201.35174 |
| [12] | Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1061.35520 |
| [13] | Benjamin, T. B., The stability theory of solitary waves, (Proc. R. Soc. London A, 328 (1972)), 153-183 |
| [14] | Craig, W., An existence theory for water waves and the Boussinesq and Korteweg-deVries scaling limits, Commun. Part. Diff. Eq., 10, 787-1003 (1985) · Zbl 0577.76030 |
| [15] | Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of a deep fluid, Zh. Prikl. Mekh. Tekh. Fiz.. Zh. Prikl. Mekh. Tekh. Fiz., J. Appl. Mech. Tech. Phys., 2, 190-194 (1968) |
| [16] | Luke, J. C., A variational principle for a fluid with a free surface, J. Fluid Mech., 27, 395-397 (1967) · Zbl 0146.23701 |
| [17] | Whitman, G. B., Variational methods and applications to water waves, (Proc. R. Soc. A, 229 (1967)), 6-25 · Zbl 0163.21104 |
| [18] | Broer, L. J.F., On the Hamiltonian theory of surface waves, Appl. Sci. Res., 30, 430-446 (1974) · Zbl 0314.76014 |
| [19] | Miles, J. W., On Hamilton’s principle for surface waves, J. Fluid Mech., 83, 153-158 (1977) · Zbl 0377.76014 |
| [20] | Miles, J. W., Hamiltonian formulations for surface waves, Appl. Sci. Res., 37, 103-110 (1981) · Zbl 0476.76025 |
| [21] | Benjamin, T. B.; Olver, P. J., Hamiltonian structure, symmetries and conservation laws for water waves, J. Fluid Mech., 125, 137-185 (1982) · Zbl 0511.76020 |
| [22] | Radder, A. C., An explicit Hamiltonian formulation of surface waves in water of finite depth, J. Fluid Mech., 237, 435-455 (1992) · Zbl 0747.76025 |
| [23] | Olver, P. J., Applications of Lie Groups to Differential Equations (1986), Springer: Springer New York · Zbl 0656.58039 |
| [24] | Goldstein, H., Classical Mechanics (1980), Addison-Wesley: Addison-Wesley Reading, Massachusetts · Zbl 0491.70001 |
| [25] | Broer, L. J.F., Approximate equations for long water waves, Appl. Sci. Res., 31, 377-395 (1975) · Zbl 0326.76017 |
| [26] | Broer, L. J.F.; van Groesen, E. W.C.; Timmers, J. M.W., Stable model equations for long water waves, Appl. Sci. Res., 32, 619-636 (1976) · Zbl 0365.76021 |
| [27] | Olver, P. J., Hamiltonian and non-Hamiltonian models for water waves, (Ciarlet, P. G.; Roseau, M., Lecture Notes in Physics 195 - Trends in Applications of Pure Mathematics to Mechanics (1984), Springer: Springer New York), 273-290 · Zbl 0583.76014 |
| [28] | Olver, P. J., Hamiltonian perturbation theory and water waves, Contemp. Math., 28, 231-249 (1984) · Zbl 0521.76018 |
| [29] | Russian Math. Surv., 23, 123-131 (1992), English translation · Zbl 0795.58042 |
| [30] | Pöschel, J., Integrability of Hamiltonian systems on Cantor sets, Commun. Pure Appl. Math., 35, 653-695 (1982) · Zbl 0542.58015 |
| [31] | Segur, H.; Kruskal, M., Nonexistence of small-amplitude breather solutions in \(φ^4\) theory, Phys. Rev. Lett., 58, 747-750 (1987) |
| [32] | Garabedian, H., Partial Differential Equations (1964), Wiley: Wiley New York · Zbl 0124.30501 |
| [33] | Groves, M. D., Hamiltonian long-wave approximations for water waves in a uniform channel, (Debnath, L., Nonlinear Dispersive Wave Systems (1992), World Scientific: World Scientific Singapore), 99-125 |
| [34] | Groves, M. D., Hamiltonian long-wave theory for water waves in a channel (1994), (In press for Q. J. Mech. Appl. Math.) · Zbl 0868.76013 |
| [35] | Groves, M. D., Theoretical aspects of gravity-capillary water waves in non-rectangular channels (1994), submitted |
| [36] | Menyuk, C. R.; Chen, H., On the Hamiltonian structure of ion-acoustic waves and water waves in channels, Phys. Fluids, 29, 998-1003 (1986) · Zbl 0593.76124 |
| [37] | Kadomtsev, B. B.; Petviashvili, V. I., On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15, 539-541 (1970) · Zbl 0217.25004 |
| [38] | Lamb, H., Hydrodynamics (1924), CUP: CUP Cambridge · JFM 50.0567.01 |
| [39] | Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley-Interscience: Wiley-Interscience New York · Zbl 0373.76001 |
| [40] | Craig, W., Water waves, Hamiltonian systems and Cauchy integrals, (Beals, M.; Melrose, R. B.; Rauch, J., Microlocal Analysis and Nonlinear Waves (1991), Springer: Springer New York), 37-45 · Zbl 0768.76006 |
| [41] | Coifman, R.; Meyer, R., Nonlinear harmonic analysis and analytic dependence, (AMS Proc. Symp. Pure Math., 43 (1985)), 71-78 · Zbl 0587.35004 |
| [42] | Craig, W.; Sulem, C., Numerical simulation of gravity waves, J. Comp. Phys., 108, 73-83 (1993) · Zbl 0778.76072 |
| [43] | Ursell, F., The long-wave paradox in the theory of water waves, (Proc. Camb. Phil. Soc., 49 (1953)), 685-694 · Zbl 0052.43107 |
| [44] | Kaup, D. J., A higher-order wave equation and the method for solving it, Prog. Theor. Phys., 54, 396-408 (1975) · Zbl 1079.37514 |
| [45] | Sachs, R. L., On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body problem, and equivalence with the AKN S hierarchy, Physica D, 30, 1-27 (1988) · Zbl 0694.35207 |
| [46] | Zufiria, J., Weakly nonlinear non-symmetric gravity waves on water of finite depth, J. Fluid Mech., 180, 371-385 (1987) · Zbl 0622.76017 |
| [47] | Zakharov, V. E.; Fadeev, L. D., The Korteweg-de Vries equation: a completely integrable Hamiltonian system, Functional Analysis and Applications, 5, 280-287 (1971) · Zbl 0257.35074 |
| [48] | Mattioli, F., Decomposition of Boussinesq equations for shallow-water waves into a set of coupled Korteweg-de Vries equations, Phys. Fluids A, 3, 2355-2359 (1991) · Zbl 0753.76026 |
| [49] | van Groesen, E.; Pudjaprasetya, S. R., Uni-directional waves over slowly varying bottom. Part I: Derivation of a KdV-type of equation, Wave Motion, 18, 345-370 (1993) · Zbl 0819.76012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
