THE EVANS FUNCTION AND GENERALIZED MELNIKOV INTEGRALS

The Evans function, E(lambda), is an analytic function whose zeros coi ncide with the eigenvalues of the operator, L, obtained by linearizing about a travelling wave. The algebraic multiplicity of the eigenvalue lambda(0) is equal to the order of the zero of E(lambda). If m is the geometric multiplicity and p is the algebraic multiplicity of the eig envalue, the term partial derivative(lambda)(p)E(lambda(0)) is shown t o be proportional to the determinant of an m x m matrix whose entries are given by the L-2 inner products of the eigenfunctions of the adjoi nt operator L and the generalized eigenfunctions of L. Perturbation e xpressions are then derived for coefficients in the Taylor expansion o f E(lambda) at lambda = lambda(0) in the circumstance that the algebra ic multiplicity of the eigenvalue decreases under perturbation. The ex pressions are used to study the eigenvalue structure for operators obt ained by linearizing about bright solitary wave solutions to perturbed nonlinear Schrodinger equations.