Solitary Wave State in the Nonlinear Kramers Equation for Self-Propelled Particles

Abstract

We study collective phenomena of self-propagating particles using the nonlinear Kramers equation. A solitary wave state appears from an instability of the spatially uniform ordered state with nonzero average velocity. Two solitary waves with different heights merge into a larger solitary wave. An approximate solution of the solitary wave is constructed using a self-consistent method. The phase transition to the solitary wave state is either first-order or second-order, depending on the control parameters.

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

“…However, the amplitude difference increases at successive collisions and only one solitary wave survives after a long time. This behavior is slightly different from the head-on collision of two solitary waves in our previous model [11], where merging occurred at the first collision. The reason for the difference is not clear; however, it is not so surprising because various phenomena such as pair annihilation, interpenetration, and the formation of a bound state occur at the head-on collision of two general dissipative solitons depending on the control parameters [15,16].…”

“…2(d). In the previous paper, we constructed a similar phase diagram for the nonlinear Kramers equation with velocity as the variable [11]. Figure 3 shows a head-on collision of two solitary waves with slightly different amplitudes at g = 2, L x = 10, α = 5, and T = 0.1.…”

“…2(d). In the previous paper, we constructed a similar phase diagram for the nonlinear Kramers equation with velocity as the variable [11].…”

“…The solitary wave state was first found in direct numerical simulations based on the Vicsek model. In a previous paper, we showed that the solitary wave state appears in the one-dimensional nonlinear Kramers equation [11]. The nonlinear Kramers equation is a time evolution equation of the probability distribution for the position and velocity of self-propelled particles.…”

Flip motion of solitary wave in an Ising-type Vicsek model

“…There are several discussions about the formation mechanism of the solitary wave state, however, it is not clearly understood yet [16]. In previous papers, we showed that the solitary wave state can appear in the nonlinear Kramers equation [17] for the probability distribution of the position and velocity of self-propelled particles. Furthermore, we proposed a simple model of one-dimensional solitary wave state, in which the linear instability of the uniform disordered state leads to the formation of a solitary wave state [18].…”