Soler model

Abstract: We study the point spectrum of the linearization at a solitary wave solution $\phi_\omega(x)e^{-\mathrm{i}\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(\psi^*\beta\psi)\beta\psi$ (known as the Soler model) We focus on the spectral stability, that is, the absence of eigenvalues with nonzero real part, in the non-relativistic limit $\omega\lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(\tau)=|\tau|^k+O(|\tau|^K)$ for $\tau\to 0$, with $0 4/n$ An important part of the stability analysis is the proof of the absence of bifurcations of nonzero-real-part eigenvalues from the embedded threshold points at $\pm 2m\mathrm{i}$ Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model, using this family to determine the multiplicity of $\pm 2\omega\mathrm{i}$ eigenvalues of the linearized operator, and the analysis of the behaviour of "nonlinear eigenvalues" (characteristic roots of holomorphic operator-valued functions)

Abstract: 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 Schrodinger equation.

Abstract: We are interested in the cubic Dirac equation with mass $m \in [0, 1]$ in two space dimensions, which is also known as the Soler model. We conduct a thorough study on this model with initial data sufficiently small in high regularity Sobolev spaces. First, we show the global existence of the model, which is uniform-in-mass. In addition, we derive a unified pointwise decay result valid for all $m \in [0, 1]$. Last but not least, we prove the cubic Dirac equations scatter linearly with an explicit scattering speed. When the mass $m=0$, we can show an improved pointwise decay result.

Abstract: We study the spectral stability of the nonlinear Dirac operator in dimension 1+1, restricting our attention to nonlinearities of the form $f(\langle\psi,\sigma_3\psi\rangle_{\mathbb{C}^2}) \sigma_3$. We obtain bounds on eigenvalues for the linearized operator around standing wave solutions of the form $e^{-i\omega t} \phi_0$. For the case of power nonlinearities $f(s)= |s|^p$, $p>0$, we obtain a range of frequencies $\omega$ such that the linearized operator has no unstable eigenvalues on the axes of the complex plane. As a crucial part of the proofs, we obtain a detailed description of the spectra of the self-adjoint blocks in the linearized operator. In particular, we show that the condition $\langle\phi_0,\sigma_3\phi_0\rangle_{\mathbb{C}^2} > 0$ characterizes groundstates analogously to the Schrodinger case.