Global front solutions converging to an asymptotic traveling wave in a time-fractional Amari model

Abstract

We study front propagation in a time-fractional Amari neural field equation with a Caputo derivative and a Heaviside firing-rate function. For a class of nonnegative even connectivity kernels and strictly decreasing C¹ front-like initial data, we reduce the Cauchy problem to a nonlinear Volterra equation for the threshold location. We prove the existence and uniqueness of a global front solution, determine the short-time behavior of its threshold location, and identify its asymptotic speed. We also construct the associated asymptotic traveling-wave profile and establish uniqueness of the speed and profile, up to translation, in the nonzero-speed case. Using a rigidity argument, we prove the asymptotic linearity of the local increments of the threshold location and obtain uniform convergence to the profile in the threshold-centered frame. Finally, we characterize the kernel-dependent tail asymptotics of the wave profile.

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BibTeX

@misc{ishii2026global,
  title = {{Global front solutions converging to an asymptotic traveling wave in a time-fractional Amari model}},
  author = {Hiroshi Ishii},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14884},
  eprint = {2609.14884},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14884},
}

RIS

TY  - UNPB
TI  - Global front solutions converging to an asymptotic traveling wave in a time-fractional Amari model
AU  - Hiroshi Ishii
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.14884
UR  - https://arxiv.org/abs/2609.14884
AB  - We study front propagation in a time-fractional Amari neural field equation with a Caputo derivative and a Heaviside firing-rate function. For a class of nonnegative even connectivity kernels and strictly decreasing C^1 front-like initial data, we reduce the Cauchy problem to a nonlinear Volterra equation for the threshold location. We prove the existence and uniqueness of a global front solution, determine the short-time behavior of its threshold location, and identify its asymptotic speed. We also construct the associated asymptotic traveling-wave profile and establish uniqueness of the speed and profile, up to translation, in the nonzero-speed case. Using a rigidity argument, we prove the asymptotic linearity of the local increments of the threshold location and obtain uniform convergence to the profile in the threshold-centered frame. Finally, we characterize the kernel-dependent tail asymptotics of the wave profile.
ER  -

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