Instability of the Standing Pulse in Skew-Gradient Systems and Its Application to FitzHugh-Nagumo Type Systems

TL;DR

The paper addresses the stability of standing pulses in skew-gradient, non-self-adjoint reaction-diffusion systems by developing a Maslov index and spectral-flow framework that links geometric pulse profiles to spectral instability. It derives a: (i) spectral-flow instability criterion via $i(w_0)=\operatorname{sf}(\mathcal{F}_\lambda,\lambda\in[0,\widehat{\lambda}])$, and (ii) a two-parameter regularization to handle translational symmetry, showing $\mu^{CLM}(\Lambda_R,E^u(\tau),\tau\in(-\infty,T])=\operatorname{sf}(\mathcal{F}_\lambda,\lambda\in[0,\widehat{\lambda}])$ for large $T$. The approach is applied to FitzHugh-Nagumo type systems, yielding concrete instability criteria: (a) a geometric condition $u'(x_0)=0$, $v''(x_0)=0$ implies instability; (b) a threshold in the inhibitor time scale $\tau$, namely $\tau>\tau_0$, guarantees instability; and (c) a complementary stability regime when $i(w_0)=0$ and $\tau<\gamma^2$. These results provide a robust topological method for predicting pulse destabilization driven by nonlinear inhibition in non-self-adjoint settings, connecting pulse geometry to spectral data through the Maslov-Spectral Flow correspondence.

Abstract

Classical results from Sturm-Liouville theory establish that the Morse index of a one-dimensional Sturm-Liouville operator defined on $\mathbb{R}$ is equal to the number of its associated conjugate points. Recent advancements by Beck et al.~\cite{BCJLM18} have extended these results to higher-dimensional Sturm-Liouville operators on $\mathbb{R}$, utilizing the Maslov index to characterize the spectral stability of nonlinear waves in multi-component systems. In this paper, we extend this framework further to non-self-adjoint settings by investigating skew-gradient reaction-diffusion systems. By utilizing the Maslov index and spectral flow, we derive an instability criterion for standing pulses. This approach bridges the gap between variational methods and the stability index in systems where the standard self-adjoint structure is absent. As a primary application, we apply our results to FitzHugh-Nagumo type systems, where the reaction terms for both the activator and inhibitor exhibit intrinsic nonlinearities. This provides a robust topological method to account for the influence of nonlinear inhibition on pulse stability in the non-self-adjoint regime.

Figures (3)

• Figure 1: Profile of a standing pulse illustrating the geometric condition $u'(x_0)=v"(x_0)=0$ for Theorem \ref{['thm:unstable by solution']}.
• Figure 2: The rectangular homotopy path in the $(\tau, \epsilon)$-plane used to establish the equivalence between the unperturbed Maslov index and the perturbed spectral flow.
• Figure 3: The rectangular homotopy path in the $(\lambda, \theta)$-plane used to evaluate the spectral flow near the origin.

Theorems & Definitions (53)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Proposition 2.1