Emergent behaviors of the kinetic Lohe Hermitian sphere model

TL;DR

This paper investigates the well-posedness and emergent behaviors of the kinetic Lohe Hermitian sphere model, a complex analogue of the Lohe particle model, providing conditions for solutions and analyzing their collective dynamics.

Contribution

It introduces a measure-valued solution framework for the kinetic LHS model and analyzes emergent behaviors based on system parameters and initial data.

Findings

Established local and global well-posedness conditions.

Analyzed the evolution of the order parameter.

Demonstrated emergent behaviors with free flows.

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.