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<p>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 <i>φ</i>, and couple it with the associated modulated perturbation <Emphasis Type="BoldItalic">V</Emphasis> along with the unmodulated perturbation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\widetilde{V}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>V</mi> <mo>~</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> to close a nonlinear iteration argument.</p>

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Nonlinear stability of large-period traveling waves bifurcating from the heteroclinic loop in the FitzHugh-Nagumo equation

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Ji Li,
  • Ke Wang,
  • Qiliang Wu,
  • Qing Yu

摘要

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 φ, and couple it with the associated modulated perturbation V along with the unmodulated perturbation \(\widetilde{V}\) V ~ to close a nonlinear iteration argument.