Symmetries of some free boundary problem in hydrodynamics
Abstract
The free boundary problem of water waves in three space dimensions without surface tension is considered. The problem consists of the Laplace equation on the velocity potential, and of the kinematic and dynamic boundary conditions. T. Brooke Benjamin and P. Olver have showed that the methods of group analysis of differential equations can be applied to such problems. The group analysis is based on finding infinitesimal symmetries inherent to the problem. The key point is that each infinitesimal symmetry generates a one-parameter group of symmetries, and that transforming a given solution of the problem by any of the symmetries produces a continuous family of other solutions. The aim of the present paper is demonstration of application of the group analysis to the problem. The base infinitesimal symmetries of the problem are deduced, their physical meanings are revealed, and their commutators are computed. It is shown that all the infinitesimal symmetries of the problem form a 13-dimensional non-solvable Lie algebra, and the corresponding Lie group of symmetries is generated by horizontal and vertical translations, time translation, variation of base-level for potential, horizontal rotations, horizontal and vetical Galilean boosts, vertical acceleration, gravity-compensated rotations, and scaling. All the results agree with the ones obtained by T. Brooke Benjamin and P. Olver.
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