Soliton dynamics in the ABS nonlinear spinor model with external fields

Abstract

We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva, Barashenkov, and Saxena [1] (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a Collective Coordinates Theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the collective coordinates. This Lagrangian differs only in the charge and mass with the Lagrangian of a collective coordinates theory for the Gross-Neveu equation. Thus the soliton dynamics in the ABS spinor model is qualitatively the same as in the Gross-Neveu equation, but quantitatively it differs. These results of the collective coordinates theory are confirmed by simulations, i.e., by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials.

Figures (10)

• Figure 1: Soliton dynamics with the ramp potential $V(x)=-V_1\,x$, for $\omega=0.9$ and $V_1=0.01$. Panel (a): soliton profiles at $t=0$ (dashed line) and at $t_f=100$ (solid line). Panels (b), (c) and (d): comparison of the simulation results for the soliton position $q(t)$, velocity $v(t)$, and momentum $P(t)$ (solid lines) with the CC results (red circles). Initial conditions: $q(0)=-46$, and $v(0)=0$.
• Figure 2: Soliton dynamics with the ramp potential $V(x)=-V_1\,x$ in the relativistic regime. Panel (a): soliton profiles at $t=0$ (dashed line) and at $t_f=4000$ (solid line). Notice that the width of the charge density is Lorentz-contracted, while the height is increased such that the charge is conserved. Panels (b), (c) and (d): comparison of the simulation results for the soliton position $q(t)$, velocity $v(t)$, and momentum $P(t)$ (solid lines) with the CC results (red circles). Initial conditions and parameters: $q(0)=-80$, $v(0)=0.9$, $\omega=0.9$, $V_1=0.01$.
• Figure 3: Soliton dynamics with the ramp potential $V(x)=-V_1\,x$, for $\omega=0.71$ and $V_1=0.01$. Panel (a): soliton profiles at $t=0$ (dashed line) and at $t_f=100$ (solid line). Note that the soliton is unstable. Panels (b), (c) and (d): comparison of the simulation results for the soliton position $q(t)$, velocity $v(t)$, and momentum $P(t)$ (solid lines) with the CC results (red circles). Initial conditions: $q(0)=-46$, $v(0)=0$.
• Figure 4: Soliton dynamics with the harmonic potential $V(x)=\frac{V_2}{2}\,x^2$, for $\omega=0.9$ and $V_2=0.0001$. Panel (a): soliton profiles at $t=0$ (dashed line) and at $t_f=800$ (solid line). Panels (b), (c) and (d): comparison of the simulation results for the soliton position $q(t)$, velocity $v(t)$, and momentum $P(t)$ (solid lines) with the CC results (red circles). Initial conditions: $q(0)=0$, $v(0)=0.1$.
• Figure 5: Soliton dynamics with the harmonic potential $V(x)=\frac{V_2}{2}\,x^2$, for $\omega=0.74$ and $V_2=0.0001$. Panel (a): soliton profiles at $t=0$ (dashed line) and at $t_f=3000$ (solid line). The soliton is clearly unstable. Panels (b), (c) and (d): comparison of the simulation results for the soliton position $q(t)$, velocity $v(t)$, and momentum $P(t)$ (solid lines) with the CC results (red circles). Initial conditions: $q(0)=0$, $v(0)=0.1$.