以运动方向为变量的非线性Kramers方程的一维和二维孤波态
One- and Two-dimensional Solitary Wave States in the Nonlinear Kramers Equation with Movement Direction as a Variable
摘要：
我们研究了自推进粒子的非线性Kramers方程的直接数值模拟。在我们的前一篇论文中，我们研究了自推进粒子的速度变量在一维。在本文中，我们考虑另一个模型，其中每个粒子表现出定向运动。移动方向用变量$ϕ$表示。我们发现，一维孤立波状态出现在直接数值模拟的非线性Kramers方程在一维和二维系统，这是我们以前的结果的推广。此外，我们发现二维局域态的情况下，每个自推进粒子表现出旋转运动。二维定域态的质心呈现圆周运动，这意味着集体旋转运动。最后，我们考虑一个简单的一维模型方程来定性地理解孤波态的形成。
英文摘要：
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 $ϕ$. 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.
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