Group foliation of equations in geophysical fluid dynamics

Pages: 1571 - 1586, Volume 27, Issue 4, August 2010

doi:10.3934/dcds.2010.27.1571       Abstract        Full Text (228.5K)       Related Articles

Jeffrey J. Early - College of Oceanic and Atmospheric Sciences, 104 COAS Admin Bldg, Oregon State University, Corvallis, OR 97331-5503, United States (email)
Juha Pohjanpelto - Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, United States (email)
Roger M. Samelson - College of Oceanic and Atmospheric Sciences, 104 COAS Admin Bldg, Oregon State University, Corvallis, OR 97331-5503, United States (email)

Abstract: The method of group foliation can be used to construct solutions to a system of partial differential equations that, as opposed to Lie's method of symmetry reduction, are not invariant under any symmetry of the equations. The classical approach is based on foliating the space of solutions into orbits of the given symmetry group action, resulting in rewriting the equations as a pair of systems, the so-called automorphic and resolvent systems, involving the differential invariants of the symmetry group, while a more modern approach utilizes a reduction process for an exterior differential system associated with the equations. In each method solutions to the reduced equations are then used to reconstruct solutions to the original equations. We present an application of the two techniques to the one-dimensional Korteweg-de Vries equation and the two-dimensional Flierl-Petviashvili (FP) equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest.

Keywords:  Symmetry, exterior differential system, group foliation, geophysical fluid dynamics.
Mathematics Subject Classification:  Primary: 34A26, 58J70; Secondary: 58A15.

Received: September 2009;      Revised: February 2010;      Published: March 2010.