Parametric spatial solitary waves

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Abstract
We consider the copropagation of three (pump, Stokes, and anti-Stokes) self-trapped beams that interact through the parametric four-photon-mixing process in a (1 + 1 dimension) uniform self-focusing nonlinear medium. The stationary propagation involving three trapped beams is shown to be possible below a critical frequency shift when the phase difference between the three waves is either matched or highly mismatched. Most importantly, the steady-state solutions are found to be stable, as in the case of presence of only one soliton beam, except that the two (in-phase) fundamental stationary states launched in parallel will not return to their original states periodically after collision; rather, they may repel each other on collision and propagate in symmetrically tilted directions thereafter with increasing separation.