Validated spectral stability via conjugate points

Abstract

Classical results from Sturm-Liouville theory state that the number of unstable eigenvalues of a scalar, second-order linear operator is equal to the number of associated conjugate points. Recent work has extended these results to a much more general setting, thus allowing for spectral stability of nonlinear waves in a variety of contexts to be determined by counting conjugate points. However, in practice, it is not yet clear whether it is easier to compute conjugate points than to just directly count unstable eigenvalues. We address this issue by developing a framework for the computation of conjugate points using validated numerics. Moreover, we apply our method to a parameter-dependent system of bistable equations and show that there exist both stable and unstable standing fronts. This application can be seen as complimentary to the classical result via Sturm-Louiville theory that in scalar reaction-diffusion equations pulses are unstable whereas fronts are stable, and to the more recent result of "Instability of pulses in gradient reaction-diffusion systems: a symplectic approach," by Beck et. al., that symmetric pulses in reaction-diffusion systems with gradient nonlinearity are also necessarily unstable.