Communications in Mathematical Sciences

Volume 21 (2023)

Number 5

Emergent behaviors of the kinetic Lohe Hermitian sphere model

Pages: 1171 – 1213

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n5.a1

Authors

Junhyeok Byeon (Research Institute of Basic Sciences, Seoul National University, Seoul, South Korea)

Seung-Yeal Ha (Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, South Korea)

Gyuyoung Hwang (Department of Mathematical Sciences, Seoul National University, Seoul, South Korea)

Hansol Park (Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada)

Abstract

We study a global well-posedness of measure-valued solutions to the kinetic Lohe Hermitian sphere (LHS) model derived from the Lohe tensor (LT) model on the set of rank‑1 complex tensors (i.e. complex vectors) with the same size and investigate emergent behaviors. The kinetic LHS model corresponds to a complex analogue of the kinetic LS model which has been extensively studied in the literature on the aggregation modeling of Lohe particles on the unit sphere in Euclidean space. In this paper, we provide several frameworks in terms of system parameters and initial data leading to the local and global well-posedness of measure-valued solutions. In particular, we show emergent behaviors of the kinetic LHS model with the same free flows by analyzing the temporal evolution of the order parameter.

Keywords

emergence, Kuramoto model, Lohe Hermitian sphere model, order parameter

2010 Mathematics Subject Classification

34D06, 70F10, 70G60

Received 9 September 2021

Received revised 29 July 2022

Accepted 14 September 2022

Published 30 August 2023