Speed-of-light pulses in the massless nonlinear Dirac equation with a potential

TL;DR

This work analyzes the massless nonlinear Dirac equation in $1+1$ dimensions with scalar-scalar self-interaction under three external potentials. Through numerical simulations and exact analysis for a constant potential, it reveals that an initially localized pulse rapidly splits into two oppositely moving pulses at the speed of light, with the nonlinear term vanishing and the dynamics reducing to the linear Dirac equation; charge and energy are preserved, and momentum conservation emerges under symmetry conditions. The exact solutions for constant potential provide analytical insight and serve as rigorous benchmarks for the observed light-speed pulse splitting, linking nonlinear dynamics to linear relativistic propagation. The findings illuminate how nonlinearity can effectively deactivate in the long-time evolution under certain external fields, with potential implications for relativistic soliton dynamics in structured media.

Abstract

We consider the massless nonlinear Dirac (NLD) equation in $1+1$ dimension with scalar-scalar self-interaction $\frac{g^2}{2} (\barΨ Ψ)^2$ in the presence of three external electromagnetic potentials $V(x)$, a potential barrier, a constant potential, and a potential well. By solving numerically the NLD equation, we find that, for all three cases, after a short transit time, the initial pulse breaks into two pulses which are solutions of the massless linear Dirac equation traveling in opposite directions with the speed of light. During this splitting the charge and the energy are conserved, whereas the momentum is conserved when the solutions possess specific symmetries. For the case of the constant potential, we derive exact analytical solutions of the massless NLD equation that are also solutions of the massless linearized Dirac equation.

Figures (7)

• Figure 1: Snapshots of the charge density. Left side: $t^{\star}=0$ (black solid lines), $t^{\star}=2$ (red dashed lines) and $t^{\star}=4$ (blue dotted lines). Right side: $t^{\star}=0$ (black solid lines), $t^{\star}=20$ (red dashed lines) and $t^{\star}=40$ (blue dotted lines). Upper panels: potential barrier $V_{1}(x)=\omega + \mu\, {\rm sech}\, 2 \beta x$, middle panels: constant potential $V_{2}(x)=\omega$, lower panels: potential well $V_{3}(x)=\omega - \mu \, {\rm sech}\, 2 \beta x$. Parameters: $g=1$, $\omega=0.9$ and $\mu=1$. IC: Eqs. (\ref{['eq5']}) with (\ref{['eq8']})-(\ref{['eq9']}).
• Figure 2: Snapshots of the real and imaginary parts of the spinor components $\psi$ and $\chi$. Black solid lines: $t^{\star}=0$, red dashed lines: $t^{\star}=2$, blue dotted lines: $t^{\star}=4$. Constant potential $V_2(x)=\omega$. Parameters and ICs: same as in Fig. \ref{['fig1']}.
• Figure 3: Same as Fig. \ref{['fig7']}, but for longer times: black solid lines: $t^{\star}=0$, red dashed lines: $t^{\star}=20$, blue dotted lines: $t^{\star}=40$.
• Figure 4: Snapshots of $f_{r}(x,t)$ (solid line) and $f_{i}(x,t)$ (dashed line) at $t=t^\star=40$. Constant potential: $V(x)=\omega=0.9$, same parameters and IC as in Fig. \ref{['fig1']}.
• Figure 5: Snapshots of the charge densities of the two spinor components $\psi$ and $\chi$, using the IC from the example Eq. (\ref{['eq14']}). Black solid lines: $t^{\star}=0$, red dashed lines: $t^{\star}=10$, and blue dotted lines: $t^{\star}=20$. Potential barrier $V_1(x)$, parameters: $\omega=0.9$, $\mu=1$.