Truncated Painlevé expansion and reduction of sine‐Poisson equation to a quadrature in collisionless cold plasma

The relativistic nonlinear self‐consistent equations for a collisionless cold plasma with stationary ions is considered. The resulting system of equations for stationary and non‐stationary equations in two‐dimensions can be reduced to the sine‐Poisson equation. The truncated Painlevé expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the travelling wave solutions of the sine‐Poisson equation describing the charge density equilibrium configuration model.