Alexopoulos, J., & de Rijk, B. (2024). Nonlinear stability of periodic wave trains in the FitzHugh-Nagumo system against fully nonlocalized perturbations. arXiv preprint arXiv:2409.17859v2.
This paper investigates the nonlinear stability of periodic traveling wave solutions, also known as wave trains, in the FitzHugh-Nagumo (FHN) system when subjected to nonlocalized perturbations. The authors aim to extend existing stability theories for wave trains in reaction-diffusion systems to encompass the FHN system, which poses challenges due to its incomplete parabolicity.
The authors employ a novel approach based on pure L∞-estimates to analyze the stability of wave trains. They utilize the inverse Laplace representation of the semigroup generated by the linearized FHN system to decompose it into high-frequency and low-frequency components. For the high-frequency part, they establish exponential decay using Neumann series expansion of the resolvent. For the low-frequency part, they leverage a novel link to the Floquet-Bloch representation and apply the Cole-Hopf transform to handle critical nonlinear terms. To address regularity issues in the quasilinear iteration scheme, the authors extend the method of nonlinear damping estimates to nonlocalized perturbations using uniformly local Sobolev norms.
The study demonstrates the Lyapunov stability of diffusively spectrally stable wave trains in the FHN system against perturbations in the Cub space (bounded and uniformly continuous functions with bounded and uniformly continuous derivatives up to the third order). The authors establish that perturbed solutions converge to a modulated wave train, with the phase modulation governed by a viscous Hamilton-Jacobi equation. They derive sharp diffusive decay rates for the perturbation and establish estimates for the phase modulation and its derivatives.
The research successfully extends the L∞-theory for nonlinear stability analysis of wave trains to the non-parabolic setting of the FHN system, demonstrating the robustness of the approach. The results highlight the crucial role of diffusive smoothing and the Cole-Hopf transform in controlling the dynamics of nonlocalized perturbations. The study provides a framework for analyzing the stability of wave trains in other semilinear dissipative systems beyond the traditional parabolic setting.
This work significantly contributes to the field of nonlinear wave stability theory by providing a rigorous mathematical framework for analyzing the stability of wave trains in the FHN system, a prototypical model for excitable media, under nonlocalized perturbations. The developed techniques and insights have broader implications for understanding the stability of patterns in various physical and biological systems modeled by semilinear dissipative equations.
The study focuses on the FHN system with specific assumptions on the parameters and spectral properties of the wave trains. Further research could explore the stability of wave trains in generalized FHN systems with different parameter regimes or under more general perturbations. Investigating the applicability of the developed framework to other non-parabolic dissipative systems with different spectral properties would be a valuable extension of this work.
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