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Group foliation of equations in geophysical fluid dynamics. (English) Zbl 1192.35147

Summary: 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.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q35 PDEs in connection with fluid mechanics
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
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