Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions
Abstract
We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimensions of the form ψ(x,t) = e−iωt ψ(x) for scalar-scalar (SS) plus vector-vector (VV) interaction with interaction Lagrangian given by LI = g2/(κ+1)[(ψ̅ ψ)κ+1 +1/p (ψ̅ γμψψ̅ γμ ψ)κ+1] where p > 0 but arbitrary otherwise. We look for solutions with 0 < ω< m where $ω, m$ are frequency and mass, respectively. We find solutions for all values of ω in this range. We compute the charge Q and the energy E for each of the solitary wave solutions and explore the region in the ($p, κ$) parameter space in which solitary wave bound states exist (i.e., for which E/Q < m). We show that for all the cases while both E and Q depend on the coupling constant g, their ratio E/Q is independent of g. We further find that in case pκ≤ 1, the charge density for all the solitary waves have only single hump while for p κ> 1 there is a transition from double to single hump and we determine it as a function of ω/m. We notice that for all p there is a transition at κ=2 in the behavior of E/Q as a function of ω which we speculate is related to the onset of instability of the solutions at κ=2. We obtain the nonrelativistic reduction of the two-parameter family to a non-relativistic modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the single hump olitary waves in the domain of validity of the modified NLSE.
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BibTeX
@misc{khare2026twoparameter,
title = {{Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions}},
author = {Avinash Khare and Fred Cooper and John F. Dawson and Avadh Saxena},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18170},
eprint = {2609.18170},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18170},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions
AU - Avinash Khare
AU - Fred Cooper
AU - John F. Dawson
AU - Avadh Saxena
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18170
UR - https://arxiv.org/abs/2609.18170
AB - We obtain exact solutions of the nonlinear Dirac equation in 1+1 dimensions of the form ψ(x,t) = e^−iωt ψ(x) for scalar-scalar (SS) plus vector-vector (VV) interaction with interaction Lagrangian given by L_I = g^2/(κ+1)[(ψ̅ ψ)^κ+1 +1/p (ψ̅ γ_μψψ̅ γ^μ ψ)^κ+1] where p > 0 but arbitrary otherwise. We look for solutions with 0 < ω< m where $ω, m$ are frequency and mass, respectively. We find solutions for all values of ω in this range. We compute the charge Q and the energy E for each of the solitary wave solutions and explore the region in the ($p, κ$) parameter space in which solitary wave bound states exist (i.e., for which E/Q < m). We show that for all the cases while both E and Q depend on the coupling constant g, their ratio E/Q is independent of g. We further find that in case pκ≤ 1, the charge density for all the solitary waves have only single hump while for p κ> 1 there is a transition from double to single hump and we determine it as a function of ω/m. We notice that for all p there is a transition at κ=2 in the behavior of E/Q as a function of ω which we speculate is related to the onset of instability of the solutions at κ=2. We obtain the nonrelativistic reduction of the two-parameter family to a non-relativistic modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the single hump olitary waves in the domain of validity of the modified NLSE.
ER -