On linear instability of solitary waves for the nonlinear Dirac equation

We consider the nonlinear Dirac equation, also known as the Soler model: i\p\sb t\psi=-i\alpha \cdot \nabla \psi+m \beta \psi-f(\psi\sp\ast \beta \psi) \beta \psi, $\psi(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, f\in C\sp 2(\R), where $\alpha_j$, $j = 1,...,n$, and $\beta$ are $N \times N$ Hermitian matrices which satisfy $\alpha_j^2=\beta^2=I_N$, $\alpha_j \beta+\beta \alpha_j=0$, $\alpha_j \alpha_k + \alpha_k \alpha_j =2 \delta_{jk} I_N$. We study the spectral stability of solitary wave solutions $\phi(x)e^{-i\omega t}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\omega\to m$, proving that if $k>2/n$, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with $\omega$ sufficiently close to $m$, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh--Schroedinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov--Kolokolov stability criterion.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1203.3859 (an earlier 1D version

Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model

On the Spectral Stability of Solitary Waves

We study the spectral stability of the solitary wave solutions to the nonlinear Dirac equations. We focus on two types of nonlinearity: the Soler type and the Coulomb type. For the Soler model, we apply the Evans function technique to explore the point spectrum of the linearized operator at a solitary wave solution to the 2D and 3D cases. For the toy Coulomb model, the solitary wave solutions are no longer SU(1, 1) symmetric. We show numerically that there are no eigenvalues near 2ωi in the nonrelativistic limit (ω . m) and the spectral stability persists in spite of the absence of SU(1, 1) symmetry