Hodge Duality and the Evans Function

Two generalisations of the Evans function, for the analysis of the linearisation about solitary waves, are shown to be equivalent. The generalisation introduced by Alexander, Gardner and Jones (1990) is based on exterior algebra and the generalisation introduced by Swinton (1992) is based on a matrix formulation and adjoint systems. In regions of the complex plane where both formulations are deened, the equivalence is geometric: we show that the formulations are dual and the duality can be made explicit using Hodge duality and the Hodge star operator. Swinton's formulation excludes potential branch points at which the Alexander, Gardner and Jones formulation is well-deened. Therefore we consider the implications of equivalence on the analytic continuation of the two formulations.