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.

Figure 1

Radial profiles of the spinor components for (left) $S=0$ solitary waves and (right) $S=1$ vortices for different values of $\omega$.

Figure 2

2D Soler model with cubic ($k=1$) nonlinearity. Dependence of the (top) imaginary and (bottom) real part of the eigenvalues with respect to $\omega$. Left (right) panels correspond to $S=0$ solitary waves ($S=1$ vortices). For the sake of clarity, we only included the values $|q|\le 2$ for the imaginary part and $|q|\le 4$ for the real part. In the former case, the imaginary part of the eigenvalues for $q=0$, $q=\pm 1$, and $q=\pm 2$ is represented by, respectively, blue, red, and black lines.

Figure 3

Left: Solitary waves in the 2D Soler model with quintic ($k=2$) nonlinearity. Dependence of the (top) imaginary and (bottom) real part of the eigenvalues with respect to $\omega$ in the same format as the previous figure. Right: Solitary waves in the 3D Soler model with cubic ($k=1$) nonlinearity. Spectrum of the linearization in the one-dimensional invariant ($q=0$) subspace, which contains the eigenvalue that is responsible for the instability for $\omega \in ({\omega }_c,1)$, with ${\omega }_c\approx 0.936$.

Figure 4

Snapshots showing the evolution of the density of an unstable $S=0$ solitary wave with $\omega =0.12$. The soliton which initially had a circular shape becomes elliptical and rotates around the center of the original solitary wave.

Figure 5

Isosurfaces for the density of an $S=1$ (left) and $S=2$ (center) vortex with $k=1$, $\omega =0.6$ and (right) an $S=0$ solitary wave with $k=2$ and $\omega =0.94$.