A Robust Numerical Method to Study Oscillatory Instability of Gap Solitary Waves

The spectral problem associated with the linearization about solitary waves of spinor systems or optical coupled mode equations supporting gap solitons is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. These problems may exhibit oscillatory instabilities where eigenvalues detach from the edges of the continuous spectrum, so called edge bifurcations. A numerical framework, based on a fast robust shooting algorithm using exterior algebra is described. The complete algorithm is robust in the sense that it does not produce spurious unstable eigenvalues. The algorithm allows to locate exactly where the unstable discrete eigenvalues detach from the continuous spectrum. Moreover, the algorithm allows for stable shooting along multi-dimensional stable and unstable manifolds. The method is illustrated by computing the stability and instability of gap solitary waves of a coupled mode model.