Solitary waves in a Two Parameter Family of Generalized Nonlinear Dirac Equations in $1+1$ Dimensions

We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimension of the form $Ψ(x,t) = Φ(x) e^{-i ωt}$ where the nonlinear interactions are a combination of vector-vector (V-V) and scalar-scalar (S-S) interactions with the interaction Lagrangian given by $L_I= \frac{g^2}{(κ+1)}(\barψ ψ)^{κ+1} -\frac{g^2}{p(κ+1)}[\barψ γ_μ ψ\barψ γ^μ ψ]^{(κ+1)/2}$. This generalizes the model of ABS (N.V. Alexeeva, I.V. Barashenkov and A. Saxena, Annals Phys. {\bf 403}, 198, (2019)) by having the arbitrary nonlinearity parameter $κ>0$ and by replacing the coefficient of the V-V interaction by the arbitrary positive parameter $p>1$ which alters the relative weights of the vector-vector and the scalar-scalar interactions. We show that the solitary wave solutions exist in the entire allowed $(κ,p)$ plane for $ω/m > 1/p^{1/(κ+1)} $, for frequency $ω$ and mass $m$. These solutions have the property that their energy divided by their charge is $\it {independent} $ of the coupling constant $g$. As $ω$ increases, there is a transition from the double humped to the single humped solitons. We discuss the regions of stability of these solutions as a function of $ω,p,κ$ using the Vakhitov-Kolokolov criterion. Finally we discuss the non-relativistic reduction of the 2-parameter family of generalized ABS models to a modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the solitary waves in the domain of validity of the modified NLSE.