Finding conservation laws for partial differential equations

Abstract

In this thesis, we discuss systematic methods of finding conservation laws for systems of partial differential equations (PDEs). We first review the direct method of finding conservation laws. In order to use the direct method, one first seeks a set of conservation law multipliers so that a linear combination of the PDEs with the multipliers will yield a divergence expression. Once a set of conservation law multipliers is determined, one proceeds to find the fluxes of the conservation law. As the solution to the problem of finding conservation law multipliers is well-understood, in this thesis we focus on constructing the fluxes assuming the knowledge of a set of conservation law multipliers. First, we derive a new method called the flux equation method and show that, in general, fluxes can be found by at most computing a line integral. We show that the homotopy integral formula is a special case of the line integral formula obtained from the flux equations. We also show how the line integral formula can be simplified in the presence of a point symmetry of the PDE system and of the set of conservation law multipliers. By examples, we illustrate that the flux equation method can derive fluxes which would be otherwise difficult to find. We also review existing known methods of finding fluxes and make comparison with the flux equation method.