Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T06:48:16.931Z Has data issue: false hasContentIssue false
  • English
  • Français

Hamiltonian structure, symmetries and conservation laws for water waves

Published online by Cambridge University Press:  20 April 2006

T. Brooke Benjamin  and
P. J. Olver
T. Brooke Benjamin
Affiliation:
Mathematical Institute, 24/29 St Giles, Oxford OX1 3LB
P. J. Olver
Affiliation:
Mathematical Institute, 24/29 St Giles, Oxford OX1 3LB Present address: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A.
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 125 , December 1982 , pp. 137 - 185
Copyright
© 1982 Cambridge University Press

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

Amick, C. J. & Toland, J. F. 1981 On solitary water-waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 9–95.Google Scholar
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, 319–361.Google Scholar
Benjamin, T. B. 1974 Lectures on nonlinear wave motion. In Nonlinear Wave Motion, Lectures in Appl. Math. vol. 15, pp. 40–58. 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. 40–58. 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, 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
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, 102–163.Google Scholar
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, 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
Gel'Fand, I. M. & Dorfman, I. Ya.1979 Hamiltonian operators and related algebraic structures. Funk. Anal. 13, 13–30.Google Scholar
Hayes, W. D. 1970 Conservation of wave action and modal wave action. Proc. R. Soc. Lond. A 320, 187–208.Google Scholar
Ibragimov, N. H. 1977 Group theoretical nature of conservation laws. Lett. Math. Phys. 1, 423–428.Google Scholar
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, 1–35.Google Scholar
Longuet-Higgins, M. S. 1974 On the mass, momentum, energy and circulation of a solitary wave. Proc. R. Soc. Lond. A 337, 1–13.Google Scholar
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157–174.Google Scholar
Longuet-Higgins, M. S. 1980a Spin and angular momentum in gravity waves. J. Fluid Mech. 97, 1–25.Google Scholar
Longuet-Higgins, M. S. 1980b On the forming of sharp corners at a free surface. Proc. R. Soc. Lond. A 371, 453–478.Google Scholar
Longuet-Higgins, M. S. 1981 On the overturning of gravity waves. Proc. R. Soc. Lond. A 376, 377–400.Google Scholar
Mccowan, J. 1891 On the solitary wave. Phil. Mag. (5) 32, 45–58.Google Scholar
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, 159–161.Google Scholar
Miles, J. W. 1977 On Hamilton's principle for surface waves. J. Fluid Mech. 83, 153–158.Google Scholar
Olver, P. J. 1977 Evolution equations possessing infinitely many symmetries. J. Math. Phys. 18, 1212–1215.Google Scholar
Olver, P. J. 1979a Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Phil. Soc. 85, 143–160.Google Scholar
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. 200–239. Springer.
Olver, P. J. 1980a On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phil. Soc. 88, 71–88.Google Scholar
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).Google Scholar
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, 175–193.Google Scholar
Stiassnie, M. & Peregrine, D. H. 1980 Shoaling of finite-amplitude surface waves on water of slowly-varying depth. J. Fluid Mech. 97, 783–805.Google Scholar
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. 226–793. 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, 86–94. (Engl. transl. J. Appl. Mech. Tech. Phys. 2, 190).Google Scholar
Zeeman, E. C. 1971 Breaking of waves. In Proc. Warwick Symposium on Differential Equations and Dynamical Systems. Lecture Notes in Mathematics, vol. 206, pp. 2–6. Springer.