2D, low-frequency, relativistic solitary waves have been found recently in 2D. Particle-in-cell simulations of ultraintense laser pulses propagating in underdense plasmas, in regimes where the laser pulse undergoes energy depletion and downshifting of its frequency. These slowly propagating, spatially localized, electromagnetic structures represent an important channel of energy conversion from the laser pulse to the plasma. Pulse energy can also be converted into fast propagating structures, associated with collimated beams of ultrarelativistic electrons. Lienard–Wiechert magnetic field structures have been observed in PIC simulations to move together with focalized ultrarelativistic electron beams in the plasma.