Kavli Affiliate: Ke Wang
| First 5 Authors: Ji Li, Ke Wang, Qiliang Wu, Qing Yu,
| Summary:
A wave front and a wave back that spontaneously connect two hyperbolic
equilibria, known as a heteroclinic wave loop, give rise to periodic waves with
arbitrarily large spatial periods through the heteroclinic bifurcation. The
nonlinear stability of these periodic waves is established in the setting of
the FitzHugh-Nagumo equation, which is a well-known reaction-diffusion model
with degenerate diffusion. First, for general systems, we give the expressions
of spectra with small modulus for linearized operators about these periodic
waves via the Lyapunov-Schmidt reduction and the Lin-Sandstede method. Second,
applying these spectral results to the FitzHugh-Nagumo equation, we establish
their diffusive spectral stability. Finally, we consider the nonlinear
stability of these periodic waves against localized perturbations. We introduce
a spatiotemporal phase modulation $varphi$, and couple it with the associated
modulated perturbation $mathbf{V}$ along with the unmodulated perturbation
$mathbf{widetilde{V}}$ to close a nonlinear iteration argument.
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