  In this article we study the existence of solutions for the Dirac systems
\begin{equation}\label{e:0.1} \left\{ \begin{array}{c}
  Pu=\frac{\partial H}{\partial v}(x,u,v) \quad\hbox{on} \ M,
  Pv=\frac{\partial H}{\partial u}(x,u,v) \quad\hbox{on} \ M,
  B_{\text{CHI}}u= B_{\text{CHI}}v=0\quad\hbox{on} \ \partial M \end{array}
\right. \end{equation} where $M$ is an $m$-dimensional compact oriented
Riemannian spin manifold with smooth boundary $\partial M$, $P$ is the Dirac
operator under the boundary condition $B_{\text{CHI}}u= B_{\text{CHI}}v=0$ on
$\partial M$, $ u,v\in C^{\infty}(M,\Sigma M)$ are spinors. Using an analytic
framework of proper products of fractional Sobolev spaces, the solutions
existence results of the coupled Dirac systems are obtained for nonlinearity
with superquadratic growth rates.
