Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media

It is demonstrated that the equation iol/J/ot + l/Jxx + K 1¢12 1/1 = 0, which describes plane self-focusing and one-dimensional self-modulation can be solved exactly by reducing it to the inverse scattering problem for a certain linear differential operator. In this case exact solutions can be obtained which describe the interaction of solitary wave packets--solitons. It is shown that the amplitude and velocity of interacting solitons do not change, whereas the phase has a discontinuity. It is also demonstrated that only paired collisions of solitons occur. The results obtained are used for assessing the nonlinear stage of development of self-modulated instability.