Existence and non-existence of asymmetrically rotating solutions to a mathematical model of self-propelled motion
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Existence and non‑existence of asymmetrically rotating solutions to a mathematical model of self‑propelled motion Mamoru Okamoto1 · Takeshi Gotoda2 · Masaharu Nagayama3 Received: 20 August 2019 / Revised: 16 April 2020 © The Author(s) 2020
Abstract Mathematical models for self-propelled motions are often utilized for understanding the mechanism of collective motions observed in biological systems. Indeed, several patterns of collective motions of camphor disks have been reported in experimental systems. In this paper, we show the existence of asymmetrically rotating solutions of a two-camphor model and give necessary conditions for their existence and non-existence. The main theorem insists that the function describing the surface tension should have a concave part so that asymmetric motions of two camphor disks appear. Our result provides a clue for the dependence between the surfactant concentration and the surface tension in the mathematical model, which is difficult to be measured in experiments. Keywords Camphor model · Asymmetrically rotating solution · Reaction-diffusion system Mathematics Subject Classification 35K57 · 35Q70
* Masaharu Nagayama [email protected] Mamoru Okamoto [email protected] Takeshi Gotoda [email protected]‑u.ac.jp 1
Graduate School of Science, Hokkaido University, Kita‑Ward, Sapporo 060‑0812, Japan
2
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464‑8602, Japan
3
Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Kita‑Ward, Sapporo 060‑0812, Japan
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1 Introduction Large numbers of independent individuals sometimes cooperate as a collection. Examples where the collection develops a function can be observed in biological systems such as flocks of birds [1], schools of fish swim [49], insect swarms, bacterial colonies [50] and cell group motions [35]. Vicsek [48] has shown the appearance of collective motions by using a particle model and, subsequently to his pioneering work, theoretical research has been conducted in terms of nonlinear physics for understanding the mechanism of collective motions performed by living organisms [36, 48, 49]. On the other hand, to study collective motions from the viewpoint of experimental systems, many researchers have utilized self-propelled materials as nonbiological systems. Examples of a simple experimental system for self-propelled materials include surfactants on particles or droplets driven by the difference of the surrounding surface tension [2, 4, 6, 10, 19, 24, 27–29, 32, 33, 41, 44, 47]. In addition, experiments to control the motions of self-propelled materials by a chemical reaction have also been reported [30, 31, 45]. For the theoretical understanding of these experimental results of self-propelled materials, a mathematical model for the experimental system was introduced and its mathematical analysis was carried out. In particular, one-dimensional motions of self-propelled ma
