Zigzag-front peak-angle convergence conjecture

Let θ\theta be the direction of a zigzag front, and let θm(t)\theta_{\mathrm{m}}(t) and θp(t)\theta_{\mathrm{p}}(t) determine the two angles forming a peak, namely θθm(t)\theta-\theta_{\mathrm{m}}(t) and θ+θp(t)\theta+\theta_{\mathrm{p}}(t). Let θ\theta_* and θ\theta^* be the angles of the two closest contact points between the Frank diagram and its convex hull, with θ\theta_* the smaller angle. Zigzag-front peak-angle conjecture. The angles of the peaks of the zigzag front caused by destabilization asymptotically approach the angles of the contact points between the Frank diagram and its convex hull; in particular,

θθm(t)θ,θ+θp(t)θ\theta-\theta_{\mathrm{m}}(t)\rightarrow\theta_*,\qquad \theta+\theta_{\mathrm{p}}(t)\rightarrow\theta^*

as tt\to\infty. This conjecture describes the asymptotic selection of the angles of destabilized zigzag fronts, and the paper reports numerical evidence supporting it.

Sources & referencesView supporting material

Primary source

Hiroshi Matano, Yoichiro Mori, Mitsunori Nara and Koya Sakakibara, “Asymptotic behavior of fronts and pulses of the bidomain model”, arXiv:2105.00169 (2021).

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