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.Comment: key words: gap solitary wave, numerical Evans function, edge
bifurcation, exterior algebra, oscillatory instability, massive Thirring
model. accepted for publication in SIAD
