School of Mathematical & Statistical Sciences Faculty Publications
Document Type
Article
Publication Date
9-1-2026
Abstract
We study the existence of pseudo-traveling waves and bump solutions for two classes of hierarchical cellular neural networks (CNNs) defined over the ring of p-adic integers Zp. The first type is a p-adic CNN described by a reaction-diffusion equation, while the second type is its quantum analog obtained via Wick rotation. The p-adic CNNs are hierarchical versions of the classical Chua-Yang CNNs; these networks have a tree-like hierarchical architecture with infinitely many cells and hidden layers. The states are governed by integro-differential equations on Zp. The p-adic traveling waves behave fundamentally differently from their Archimedean counterparts. A traveling wave restricted to a p-adic sphere yields a countably infinite collection of independent patterns. We introduce the notion of pseudo-traveling waves as finite truncations of this structure and prove their existence for both the classical and quantum networks. We further establish the existence of time-independent solutions (bumps) for both models. Our theoretical results are complemented by numerical simulations that approximate pseudo-traveling-wave solutions for quantum CNNs.
Recommended Citation
Zúñiga-Galindo, W. A., B. A. Zambrano-Luna, and Chayapuntika Indoung. “Pseudo-Traveling Waves and Bumps in Quantum and Classical Hierarchical Cellular Neural Networks.” Journal of Pseudo-Differential Operators and Applications 17, no. 3 (2026): 65. https://doi.org/10.1007/s11868-026-00818-7.
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Publication Title
Journal of Pseudo Differential Operators and Applications
DOI
10.1007/s11868-026-00818-7
Comments
This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it.