One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable

TL;DR

This paper investigates solitary wave and localized states in a nonlinear Kramers equation modeling self-propelled particles with directional movement, revealing new two-dimensional localized rotating states and providing a simplified model for their formation.

Contribution

It extends previous work by analyzing directional movement in self-propelled particles, discovering two-dimensional localized rotating states, and proposing a simplified model for solitary wave formation.

Findings

One-dimensional solitary wave states observed in simulations.

Two-dimensionally localized states with rotational motion identified.

Center of mass exhibits circular motion indicating collective rotation.

Abstract

We study self-propelled particles by direct numerical simulation of the nonlinear Kramers equation for self-propelled particles. In our previous paper, we studied self-propelled particles with velocity variables in one dimension. In this paper, we consider another model in which each particle exhibits directional motion. The movement direction is expressed with a variable $\phi$. We show that one-dimensional solitary wave states appear in direct numerical simulations of the nonlinear Kramers equation in one- and two-dimensional systems, which is a generalization of our previous result. Furthermore, we find two-dimensionally localized states in the case that each self-propelled particle exhibits rotational motion. The center of mass of the two-dimensionally localized state exhibits circular motion, which implies collective rotating motion. Finally, we consider a simple one-dimensional model equation to qualitatively understand the formation of the solitary wave state.