Nonrelativistic asymptotics of solitary waves in the Dirac equation with the Soler-type nonlinearity

We use the perturbation theory to build solitary wave solutions $\phi_\omega(x)e^{-i\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\bar\psi\psi)\beta\psi$, with $f(\tau)=|\tau|^k+o(|\tau|^k)$, $k>0$, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit $\omega\lesssim m$; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schr\"odinger charge-critical, one has $Q'(\omega)<0$ for $\omega\lesssim m$, implying the absence of the degeneracy of zero eigenvalue of the linearization at a solitary wave.