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

2025/04/17 by Avinash Khare, Fred Cooper, Khare, Avinash +5 · 1 voice · 1 citation
Physics and Astronomy · #FOS: Physical sciences #Pattern Formation and Solitons (nlin.PS) #nlin.PS

paper · pdf · doi:10.48550/arxiv.2504.13299

Abstract

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= (g²)/((κ+1))(ψ ψ)^κ+1 -(g²)/(p(κ+1))[ψ γ_μ ψψ 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.

Citations

• Stability of standing periodic waves in the massive Thirring model

Cited by

• Generalized PT-symmetric nonlinear Dirac equation: exact solitary waves solutions, stability and conservation laws

Discussions

• Avinash Khare, Fred Cooper, John F. Dawson, Avadh Saxena: Solitary waves in a Two Parameter Family of Generalized Nonlinear Dirac Equations in $1+1$ Dimensions https://arxiv.org/abs/2504.13299 https:/ [bsky, 0 points, 1 comments]

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