Electron Holes in a $\kappa$ Distribution Background with Singularities

The pseudo-potential method is applied to derive diverse propagating electron hole structures, in a nonthermal or $\kappa$ particle distribution function background. The associated distribution function Ansatz reproduces the Schamel distribution of \cite{Schamel2015} in the Maxwellian ($\kappa \rightarrow \infty$) limit, providing a significant generalization of it for plasmas where superthermal electrons are ubiquitous, such as space plasmas. The pseudo-potential and the nonlinear dispersion relation are evaluated. The role of the spectral index $\kappa$ on the nonlinear dispersion relation is investigated, in what concerns the wave amplitude for instance. The energy-like first integral from Poisson's equation is applied to analyze the properties of diverse classes of solutions: with the absence of trapped electrons, with a non-analytic distribution of trapped electrons, or with a surplus of trapped electrons. Special attention is therefore paid to the non-orthodox case where the electrons distribution function exhibits strong singularities, being discontinuous or non-analytic.