  We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with
scalar-scalar self interaction $\frac{g^2}{k+1} ({\bar \Psi} \Psi)^{k+1}$, as
well as a vector-vector self interaction $\frac{g^2}{k+1} ({\bar \Psi}
\gamma_\mu \Psi \bPsi \gamma^\mu \Psi)^{\frac{1}{2}(k+1)}$. We find the exact
analytic form for solitary waves for arbitrary $k$ and find that they are a
generalization of the exact solutions for the nonlinear Schr\"odinger equation
(NLSE) and reduce to these solutions in a well defined nonrelativistic limit.
We perform the nonrelativistic reduction and find the $1/2m$ correction to the
NLSE, valid when $|\omega-m |\ll 2m$, where $\omega$ is the frequency of the
solitary wave in the rest frame. We discuss the stability and blowup of
solitary waves assuming the modified NLSE is valid and find that they should be
stable for $k < 2$.
