Department of Applied Physics, EPS, Virgen de África 7, 41011, Sevilla, Spain and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain.
Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany.
Phys Rev E. 2017 Nov;96(5-1):052219. doi: 10.1103/PhysRevE.96.052219. Epub 2017 Nov 22.
We consider the massless nonlinear Dirac (NLD) equation in 1+1 dimension with scalar-scalar self-interaction g^{2}/2(Ψ[over ¯]Ψ)^{2} in the presence of three external electromagnetic real potentials V(x), a potential barrier, a constant potential, and a potential well. By solving numerically the NLD equation, we find different scenarios depending on initial conditions, namely, propagation of the initial pulse along one direction, splitting of the initial pulse into two pulses traveling in opposite directions, and focusing of two initial pulses followed by a splitting. For all considered cases, the final waves travel with the speed of light and are solutions of the massless linear Dirac equation. During these processes 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. Decay or growth of the initial pulse is also predicted from the evolution of the charge for the case of a non-zero imaginary part of the potential.
我们考虑了在 1+1 维中存在标量-标量自相互作用 g^{2}/2(Ψ[over ¯]Ψ)^{2}的无质量非线性狄拉克（NLD）方程，以及三个外部电磁实势 V(x)、势垒、常势能和势阱。通过数值求解 NLD 方程，我们根据初始条件发现了不同的情况，即初始脉冲沿一个方向传播、初始脉冲分裂成两个相反方向传播的脉冲，以及两个初始脉冲聚焦后分裂。对于所有考虑的情况，最终波以光速传播，并且是无质量线性狄拉克方程的解。在这些过程中，电荷和能量是守恒的，而当解具有特定的对称性时，动量也是守恒的。对于常势能的情况，我们推导出了无质量 NLD 方程的精确解析解，它们也是无质量线性化狄拉克方程的解。对于势的虚部不为零的情况，从电荷的演化可以预测初始脉冲的衰减或增长。
