Topological Solitons/Numerical Analysis Seminar: Detection of Instability and Edge Bifurcation of Solitary Waves
21 February 2006 13:05 in CM221
"The spectral problem associated with the linearization about solitary waves can be formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. In the talk, a numerical framework for finding the zeros of this function, based on a fast robust shooting algorithm on exterior algebra spaces is described. The algorithm has several features, including a rigorous numerical algorithm for choosing starting values, a method for numerical analytic continuation of starting vectors, and the role of the Hodge star operator deducing a range of numerically computable forms for the Evans function. The complete algorithm is robust in the sense that it does not produce spurious unstable eigenvalues. Two illustrations of the algorithm will be presented. The first one is the computation of the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity. The instabilities in this equation are due to real eigenvalues. The second illustration is the detection of oscillatory instabilities where eigenvalues detach from the edges of the continuous spectrum, so called edge bifurcations. The algorithm allows to locate exactly where the unstable discrete eigenvalues detach from the continuous spectrum. The method is illustrated by computing the stability and instability of gap solitary waves of a coupled mode model.
[Joint work with Georg Gottwald and Tom Bridges] "