We study five relativistic plane nonlinear waves: circularly polarized waves and electrostatic plasma oscillations propagating parallel to the magnetic field, relativistic Alfvén waves, linearly polarized transverse waves propagating in zero magnetic field, and finally the relativistic analogue of the extraordinary mode propagating at an arbitrary angle to the magnetic field. When the ions are driven relativistic, they behave like electrons, and the assumption of an ‘electron–positron’ plasma guides us to equations which have the form of a one dimensional potential well. Our solutions indicate that a large-amplitude super luminous wave determines the average plasma properties, and not vice versa. For example, linearly polarized waves impose a plasma number flux equal to the relativistic addition of Nc/β and NVE, where N is the density, c the speed of light, β (>1) the ratio of the phase speed to c and VE the E × B speed measured in the frame moving with speed c/β with respect to the frame in which the phase speed is measured. The implications for cosmic ray acceleration in pulsar magnetospheres are considered.