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Anchored spirals in the driven curvature flow approximation
We study existence, asymptotics, and stability of spiral waves in a driven curvature approximation, supplemented with an anchoring condition on a circle of finite radius. We analyze the motion of curves written as graphs in polar coordinates, finding spiral waves as rigidly rotating shapes. The existence analysis reduces to a planar ODE and asymptotics are given through center manifold expansions. In the limit of a large core, we find rotation frequencies and corrections starting form a problem without curvature corrections. Finally, we demonstrate orbital stability of spiral waves by exploiting a comparison principle inherent to curvature driven flow.
@article{li2024,
title={Anchored spirals in the driven curvature flow approximation},
author={Nan Li and Arnd Scheel},
year={2024},
eprint={2312.07809},
archivePrefix={arXiv},
primaryClass={nlin.PS},
url={https://arxiv.org/abs/2312.07809},
} 
Counting compatible indexing systems for $C_{p^n}$
We count the number of compatible pairs of indexing systems for the cyclic group $C_{p^n}$. Building on work of Balchin--Barnes--Roitzheim, we show that this sequence of natural numbers is another family of Fuss--Catalan numbers. We count this two different ways: showing how the conditions of compatibility give natural recursive formulas for the number of admissible sets and using an enumeration of ways to extend indexing systems by conceptually simpler pieces.
@article{hill2022,
title={Counting compatible indexing systems for
$C_{p^n}$},
author={Michael A. Hill and Jiayun Meng and Nan Li},
journal={Orbita Mathematicae},
year={2022},
}