[en] A small-amplitude slow ion acoustic monotonic double layer in an unmagnetized plasma consisting of relativistic drifting cold electrons and nonrelativistic drifting thermal ions is investigated. By using the reductive perturbation method, Schamel-Korteweg-de Vries (SKdV) and Schamel equations are derived. We used the linearization transformation to obtained the solutions of the SKdV and Schamel equations. The method is based upon a linearization principle that can be applied on nonlinearities which have a polynomial form. We illustrate the potential of the method by finding solutions of the SKdV and Schamel equations. Furthermore, we show that the monotonic double-layer solution is a nonlinear extension of the slow ion acoustic solitary hole having a negative trapping parameter in a semi relativistic plasma.

[en] The relativistic nonlinear self-consistent equations for a collisionless cold plasma with stationary ions [R. S. Ibrahim, IMA J. Appl. Math. 68, 523 (2003)] are extended to 3 and 3+1 dimensions. The resulting system of equations is reduced to the sine-Poisson equation. The truncated Painleve expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the traveling wave solutions of the sine-Poisson equation for stationary and nonstationary equations in 3 and 3+1 dimensions describing the charge-density equilibrium configuration model