Hamiltonian Structure, Symmetries and Conservation Laws for Water Waves.
Abstract
An investigation on novel lines is made into the problem of water waves according to the perfect-fluid model, with referencce 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 and a brief discussion of implications that the present findings may carry for the approximate mathematical modelling of water waves. Details of the hydrodynamic problem are recalled, then 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 Hamiltonian evolution. Complete symmetry groups are given for several versions of the water-wave problem--easily understood forms of the main results are listed and the systematic derivations of them are explained. Conservation laws implied by the one-parameter subgroups of the full symmetry groups are worked out and a recent extension of Noether's theorem is applied relying on the Hamiltonian structure of the problem. The physical meanings of the conservation laws are examined fully and various new insights into the water-wave problem are presented.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1981
- Accession Number
- ADA110475
Entities
People
- P. J. Olver
- T. Brooke Benjamin
Organizations
- University of Wisconsin–Madison