Solitary waves in the complementary generalized ABS model

We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimension of the form ψ(x,t) =ψ(x) e- i ω t where the nonlinear interactions are a combination of vector-vector (VV) and scalar-scalar (SS) interactions with the interaction Lagrangian given by LI = g²/(κ+1)[ψ̄ γμψ ψ̄ γμ ψ](κ+1)/2 - g²/q(κ+1) (ψ̄ ψ)κ+1. This is the complement of the generalization of the Alexeeva, Barashenkov, Saxena (ABS) model {abs} that we recently studied {ak} and called the generalized ABS (i.e. gABS) model. We find that in the complementary gABS model the energy of the solitary wave divided by its charge is { independent} of the coupling constant g in accordance to what we found in the gABS model. However, unlike the solitary wave behavior in the gABS model, in this complementary gABS model all the solitary waves have a single hump, any value of 0 < ω < m is allowed and furthermore, the solitary wave bound states exist only in case κ ≤ κc, where κc depends on the value of q. Here ω and m denote frequency and mass respectively. We discuss the non-relativistic reduction of this complementary gABS model to a modified nonlinear Schr"odinger equation (NLSE) and discuss the stability of the solitary waves in the domain of validity of the modified NLSE.