Solitary Wave Solutions for the Nonlinear Dirac Equations

In this paper we prove the existence and local uniqueness of stationary states for the nonlinear Dirac equation $$i \sum_{j=0}^{3} \ga^j \pd_j \psi - m\psi + F(\bar{\psi}\psi)\psi =0$$ where $m >0$ and $F(s) = |s|^{\theta}$ for $1\leq \theta < 2.$ More precisely we show that there exists $\e_0 > 0$ such that for $\omega \in(m - \e_0, m),$ there exists a solution $\psi(t,x) = e^{-i\omega t}\phi_{\omega}(x), x_0 = t, x = (x_1, x_2, x_3),$ and the mapping from $\omega$ to $\phi_{\omega}$ is continuous. We prove this result by relating the stationary solutions to the ground states of nonlinear Schr\"{o}dinger equations.