The Maslov index, degenerate crossings and the stability of pulse solutions to the Swift-Hohenberg equation

TL;DR

This work develops a Maslov-index framework to assess spectral stability of symmetric pulse solutions in the Swift-Hohenberg equation by relating unstable eigenvalues to conjugate points of a path of Lagrangian subspaces. It generalizes the crossing-form theory to degenerate (nonregular) crossings by introducing higher-order quadratic forms whose signs determine contributions to the Maslov index, and it links these crossings to the motion of eigenvalues. A practical numerical method is introduced to compute conjugate points and unstable eigenvalues for representative pulses, including both non-snaking and snaking parameter regimes, illustrating consistency between the Maslov-count and spectral computations. The results provide a topological, parameter-robust approach to counting unstable modes and lay groundwork for validated numerics and extensions to other fourth-order Hamiltonian systems, with potential connections to classification schemes for pulse solutions. The approach has potential impact in rigorously establishing stability properties of localized structures in pattern-forming PDEs and guiding numerical studies where traditional spectral methods are challenging.

Abstract

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.

Figures (16)

• Figure 1: Example symmetric pulse profiles.
• Figure 2: The function $f(t) = \pm t^j$ near $t = 0$. The case that $f^{(j)}(0) > 0$ is depicted by the dashed line and the case that $f^{(j)}(0)< 0$ is given by the solid line.
• Figure 3: Representing $\ell(t)$ as a graph for $t$ near $t_0$.
• Figure 4: The motion of the eigenvalues of $B(t)$ within $\epsilon$ of $0$ in the case that $R^{(2)}(u)$ is the lowest order non-vanishing quadratic form.
• Figure 5: The motion of the eigenvalues of $B(t)$ within $\epsilon$ of $0$ in the case that $R^{(3)}(u)$ is the lowest order non-vanishing quadratic form.

Theorems & Definitions (79)

• Remark 1.3
• Lemma 1.4
• proof
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Definition 2.1
• Theorem 2.2
• Remark 2.3
• Definition 2.4