Abstract
Todd Kapitula
Title: Edge bifurcations for near integrable systems
Abstract: When studying the linear stability of waves for near integrable
systems, a fundamental problem is the location of the point spectrum for
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 initially not
localized. This phenomenon is also known as an {\it 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 the solutions to
the Lax pair underlying the integrable problem play in the construction of
the Evans function and the detection of edge bifurcations. Using the
Zakharov-Shabat eigenvalue problem as an illustration, we show the
connection between the inverse scattering formalism and linear stability
analysis of waves. In particular, we show a direct connection between the
scattering coefficients and the Evans function.
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