Group foliation of equations in geophysical fluid dynamics

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 invari- ant 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 reduc- tion 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 equa- tions. 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 analyt- ical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest. 1