Self-accelerating self-trapped nonlinear beams of Maxwell's equations

We present shape-preserving self-accelerating beams of Maxwell's equations with optical nonlinearities. Such beams are exact solutions to Maxwell's equations with Kerr or saturable nonlinearity. The nonlinearity contributes to self-trapping and causes backscattering. Those effectstogether with diffraction effectswork to maintain shape-preserving acceleration of the beam on a circular trajectory. The backscattered beam is found to be a key issue in the dynamics of such highly non-paraxial nonlinear beams. To study thatwe develop two new techniques: projection operator separating the forward and backward wavesand reverse simulation. Finallywe discuss the possibility that such beams would reflect themselves through the nonlinear effectto complete a 'U' shaped trajectory.