Solitary Wave Solutions for the Nonlinear Dirac Equations

Abstract

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

18 pages