Solitary waves in the complementary generalized ABS model

TL;DR

This paper derives exact solitary wave solutions for a nonlinear Dirac equation with combined vector and scalar interactions, analyzing their properties, stability, and non-relativistic limits in a novel generalized ABS model.

Contribution

It introduces the complementary gABS model with exact solutions, explores their stability, and connects the relativistic solutions to a modified nonlinear Schrödinger equation.

Findings

Solitary waves exist for all (κ, q) in the model.

Energy per charge of solitary waves is independent of coupling g.

Bound states exist only for κ ≤ κ_c, depending on q.

Abstract

We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimension of the form $ \Psi(x,t) =\Phi(x) \rme^{-\rmi \omega t}$ where the nonlinear interactions are a combination of vector-vector and scalar-scalar interactions with the interaction Lagrangian given by $L_I = \frac{g^2}{(\kappa+1)}[\bar{\psi} \gamma_{\mu}\psi \bar{\psi} \gamma^{\mu} \psi]^{(\kappa+1)/2} - \frac{g^2}{q(\kappa+1)}(\bar{\psi} \psi)^{\kappa+1}$, where $\kappa>0$ and $q>1$. This is the complement of the generalization of the ABS model \cite{abs} that we recently studied \cite{ak} and denoted as the gABS model. We show that like the gABS model, in the complementary gABS models the solitary wave solutions also exist in the entire $(\kappa, q)$ plane and further in both models energy of the solitary wave divided by its charge is {\it independent} of the coupling constant $g$. However, unlike the gABS model here all the solitary waves are single humped, any value of $0 < \omega < m$ is allowed and further unlike the gABS model, for this complementary gABS model the solitary wave bound states exist only in case $\kappa \le \kappa_c$, where $\kappa_c$ depends on the value of $q$. Here $\omega$ and $m$ denote frequency and mass, respectively. We discuss the regions of stability of these solutions as a function of $\omega,q,\kappa$ using the Vakhitov-Kolokolov criterion. Finally we discuss the non-relativistic reduction of the two-parameter family of this complementary generalized ABS 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.