Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

Stability of nonlinear Dirac solitons under the action of external potential

The instabilities observed in direct numerical simulations of the Gross-Neveu equation under linear and harmonic potentials are studied. The Lakoba algorithm, based on the method of characteristics, is performed to numerically obtain the two spinor components. We identify non-conservation of energy and charge in simulations with instabilities, and we find that all studied solitons are numerically stable, except the low-frequency solitons oscillating in the harmonic potential over long periods of time. These instabilities, as in the case of the Gross-Neveu equation without potential, can be removed by imposing absorbing boundary conditions. The dynamics of the soliton is in perfect agreement with the prediction obtained using an Ansatz with only two collective coordinates, namely, the position and momentum of the center of mass. We employ the temporal variation of both field energy and momentum to determine the evolution equations satisfied by the collective coordinates. By applying the same methodology, we also demonstrate the spurious character of the reported instabilities in the Alexeeva-Barashenkov-Saxena model under external potentials.

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

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.

Stability of Solitary Waves and Vortices in a 2D Nonlinear Dirac Model

We explore a prototypical two-dimensional massive model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis, illustrating the potential of spinor solutions to be neutrally stable in a wide parametric interval of frequencies. Solutions of higher vorticity are generically unstable and split into lower charge vortices in a way that preserves the total vorticity. These conclusions are found not to be restricted to the case of cubic two-dimensional nonlinearities but are found to be extended to the case of quintic nonlinearity, as well as to that of three spatial dimensions. Our results also reveal nontrivial differences with respect to the better understood nonrelativistic analogue of the model, namely the nonlinear Schrödinger equation.