Edge Bifurcations for Near Integrable Systems via Evans Function Techniques

When studying the linearstability of waves for near integrable systems, a fundamental problem is the location of the point spectrum of the linearized operator. Internal modes may be created upon the perturbation, i.e., eigenvalues may bifurcate out of the continuous spectrum, even if the corresponding eigenfunction is not initially localized. This phenomenon is also known as an edge bifurcation. It has recently been shown that the Evans function is a powerful tool when one wishes to detect an edge bifurcation and track the resulting eigenvalues. It has been an open question as to the role played by the solutions to the Lax pair, associated with the integrable problem, in the construction of the Evans function and the detection of edge bifurcations. Using the Zakharov--Shabat eigenvalue problem and the massive Thirring model as illustrations, we show the connection between the inverse scattering formalism and the linear stability analysis of waves. In particular, we show a direct connection between the scattering coefficients and the Evans function. Last, using perturbations of the massive Thirring model, we show how the Evans function can be used to predict the location of bifurcating edge eigenvalues.