Accelerating self-modulated nonlinear waves in weakly and strongly magnetized relativistic plasmas

It is known that a nonlinear Schr\"odinger equation describes the self-modulation of a large amplitude circularly polarized wave in relativistic electron-positron plasmas in the weakly and strongly magnetized limits. Such equation can be written as a modified second Painlev\'e equation, producing accelerated propagating wave solutions for those nonlinear plasmas. The acceleration depends on the plasma magnetization. This is different to the usual soliton solution propagating at constant speed