The Maslov Index, Degenerate Crossings and the Stability of Pulse Solutions to the Swift-Hohenberg equation

In the scalar Swift-Hohenberg equation with quadratic-cubic nonlinearity, it is known that symmetric pulse solutions exist for certain parameter regions. In this paper we develop a method to determine the spectral stability of these solutions by associating a Maslov index to them. This requires extending the method of computing the Maslov index introduced by Robbin and Salamon [Topology 32, no.4 (1993): 827-844] to so-called degenerate crossings. We extend their formulation of the Maslov index to degenerate crossings of general order in the case where the intersection is fully degenerate, meaning that if the dimension of the intersection is k, then each of the k crossings is a degenerate one. We then argue that, in this case, this index coincides with the number of unstable eigenvalues for the linearized evolution equation. Furthermore, we develop a numerical method to compute the Maslov index associated to symmetric pulse solutions. Finally, we consider several solutions to the Swift-Hohenberg equation and use our method to characterize their stability.