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Hamiltonian structure, symmetries and conservation laws for water waves

  • T. Brooke Benjamin (a1) and P. J. Olver (a1) (a2)
Abstract

An investigation on novel lines is made into the problem of water waves according to the perfect-fluid model, with reference to wave motions in both two and three space dimensions and with allowance for surface tension. Attention to the Hamiltonian structure of the complete nonlinear problem and the use of methods based on infinitesimal-transformation theory provide a Systematic account of symmetries inherent to the problem and of corresponding conservation laws.

The introduction includes an outline of relevant elements from Hamiltonian theory (§ 1.1) and a brief discussion of implications that the present findings may carry for the approximate mathematical modelling of water waves (§1.2). Details of the hydrodynamic problem are recalled in §2. Then in §3 questions about the regularity of solutions are put in perspective, and a general interpretation is expounded regarding the phenomenon of wave-breaking as the termination of smooth Hamil- tonian evolution. In §4 complete symmetry groups are given for several versions of the water-wave problem : easily understood forms of the main results are listed first in §4.1, and the systematic derivations of them are explained in §4.2. Conservation laws implied by the one-parameter subgroups of the full symmetry groups are worked out in §5, where a recent extension of Noether's theorem is applied relying on the Hamiltonian structure of the problem. The physical meanings of the conservation laws revealed in §5, to an extent abstractly there, are examined fully in §6 and various new insights into the water-wave problem are presented.

In Appendix 1 the parameterized version of the problem is considered, covering cases where the elevation of the free surface is not a single-valued function of horizontal position. I n Appendix 2 a general method for finding the symmetry groups of free-boundary problems is explained, and the exposition includes the mathematical material underlying the particular applications in §§4 and 5.

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References
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Amick, C. J. & Toland, J. F. 1981 On solitary water-waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 995.
Arnold, V. I. 1966 Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydromécanique des fluides parfaits. Ann. Inst. Fourier Grenoble 16, 319361.
Benjamin, T. B. 1974 Lectures on nonlinear wave motion. In Nonlinear Wave Motion, Lectures in Appl. Math. vol. 15, pp. 4058. American Mathematical Society.
Benjamin, T. B. 1980 Theoretical problems posed by gravity-capillary waves with edge constraints. In Trends in Applications of Pure Mathematics to Mechanics II (ed. H. Zorski), pp. 4058. Pitman.
Benjamin, T. B., Bona, J. E. & Mahony, J. J. 1972 Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. Lond. A 272, 4778.
Benjamin, T. B. & Mahony, J. J. 1971 On an invariant property of water waves. J. Fluid Mech. 49, 385389.
Bluman, G. W. & Cole, J. D. 1974 Similarity Methods for Differential Equations. Appl. Math. Sci., vol. 13. Springer.
Ebin, D. G. & Marsden, J. 1970 Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102163.
Eisenhart, L. P. 1933 Continuous Groups of Transformations. Princeton University Press.
Friedrichs, K. O. & Hyers, D. H. 1954 The existence of solitary waves. Comm. Pure Appl. Math. 7, 517550.
Gardner, C. S. 1971 Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys. 12, 15481551.
Gel'Fand, I. M. & Dorfman, I. Ya.1979 Hamiltonian operators and related algebraic structures. Funk. Anal. 13, 1330.
Hayes, W. D. 1970 Conservation of wave action and modal wave action. Proc. R. Soc. Lond. A 320, 187208.
Ibragimov, N. H. 1977 Group theoretical nature of conservation laws. Lett. Math. Phys. 1, 423428.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press. (Dover reprint 1945.)
Lax, P. D. 1978 A Hamiltonian approach to the KdV and other equations. In Group Theoretic Methods in Physics, 5th Int. Colloq. Academic.
Longuet-Higgins, M. S. 1950 A theory of the origin of microseisms. Phil. Trans. R. Soc. Lond. A 243, 135.
Longuet-Higgins, M. S. 1974 On the mass, momentum, energy and circulation of a solitary wave. Proc. R. Soc. Lond. A 337, 113.
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157174.
Longuet-Higgins, M. S. 1980a Spin and angular momentum in gravity waves. J. Fluid Mech. 97, 125.
Longuet-Higgins, M. S. 1980b On the forming of sharp corners at a free surface. Proc. R. Soc. Lond. A 371, 453478.
Longuet-Higgins, M. S. 1981 On the overturning of gravity waves. Proc. R. Soc. Lond. A 376, 377400.
Mccowan, J. 1891 On the solitary wave. Phil. Mag. (5) 32, 4558.
Marsden, J. 1974 Applications of Global Analysis in Mathematical Physics. Publish or Perish.
Milder, D. M. 1977 A note regarding ‘On Hamilton's principle for surface waves’. J. Fluid Mech. 83, 159161.
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153158.
Olver, P. J. 1977 Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18, 12121215.
Olver, P. J. 1979a Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Phil. Soc. 85, 143160.
Olver, P. J. 1979b How to find the symmetry group of a differential equation. Appendix in D. H. Sattinger, Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Mathematics, vol. 762, pp. 200239. Springer.
Olver, P. J. 1980a On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phil. Soc. 88, 7188.
Olver, P. J. 1980b Applications of Lie Groups to Differential Equations. Mathematical Institute, University of Oxford, Lecture Notes.
Olver, P. J. 1982 Conservation laws of free boundary problems and the classification of conservation laws for water waves. Trans. Am. Math. Soc. (to appear).
Ovsiannikov, L. V. 1982 Group Analysis of Differential Equations (translated by W. F. Ames). Academic.
Starr, V. T. 1947 Momentum and energy integrals for gravity waves of finite height. J. Mar. Res. 6, 175193.
Stiassnie, M. & Peregrine, D. H. 1980 Shoaling of finite-amplitude surface waves on water of slowly-varying depth. J. Fluid Mech. 97, 783805.
Thom, R. 1975 Structural Stability and Morphogenesis. Benjamin.
Truesdell, C. & Toupin, R. A. 1960 The classical field theories. In Handbuch der Physik, vol. III/1, pp. 226793. Springer.
Whittaker, E. T. 1937 A treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th edn. Cambridge University Press.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Fiz. 9, 8694. (Engl. transl. J. Appl. Mech. Tech. Phys. 2, 190).
Zeeman, E. C. 1971 Breaking of waves. In Proc. Warwick Symposium on Differential Equations and Dynamical Systems. Lecture Notes in Mathematics, vol. 206, pp. 26. Springer.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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