Group foliation of equations in geophysical fluid dynamics

Jeffrey J Early,Roger M Samelson,Juha Pohjanpelto

doi:10.3934/dcds.2010.27.1571

Jeffrey J Early, Roger M Samelson + Show 1 more

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https://doi.org/10.3934/dcds.2010.27.1571

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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.

Highlights

The main theoretical objects of study in geophysical fluid dynamics are systems of partial differential equations describing the time evolution and thermodynamics of rotating stratified fluids [23, 28]
Finding exact analytical solutions to these nonlinear partial differential equations (PDEs) can be exceptionally difficult, and increasingly physicists have instead turned to numerical solution methods
In the case of the Korteweg-de Vries (KdV) equation (4), the general solution to the third order equation is recovered from the solutions of the Abel equation of the second kind, a first order ODE

Summary

Introduction

The main theoretical objects of study in geophysical fluid dynamics are systems of partial differential equations describing the time evolution and thermodynamics of rotating stratified fluids [23, 28]. Finding exact analytical solutions to these nonlinear partial differential equations (PDEs) can be exceptionally difficult, and increasingly physicists have instead turned to numerical solution methods. The rapid growth of computing power in the last few decades has made. Exterior differential system, group foliation, geophysical fluid dynamics.

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