{
  "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 $$L_{I} = \\frac{g^2}{κ+1}[(\\barψ ψ)^{κ+1} +\\frac{1}{p} (\\barψ γ_μψ\\barψ γ^μ ψ)^{κ+1}]$$ 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 $pκ\\le 1$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18170",
  "author_total": 4,
  "authors": [
    {
      "name": "Avinash Khare"
    },
    {
      "name": "Fred Cooper"
    },
    {
      "name": "John F. Dawson"
    },
    {
      "name": "Avadh Saxena"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ccw7rk15959w6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18170",
  "fulltext_url": null,
  "id": "ccw7rk15959w6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18170",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "nlin.PS"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18170",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ccw7rk15959w6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.18170",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18170"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.PS",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions",
  "title_text": "Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/ccw7rk15959w6",
  "venue": "arXiv",
  "versions": []
}