Integrals of motion of a nonlinear four-photon interaction

The results are reported of an analysis of a system of equations describing a four-photon interaction between electromagnetic wave beams in a nonlinear (cubic) medium. The integrals of motion of the system are found and are then used to draw conclusions on the qualitative behavior of the interacting fields. Equations are derived for the change in the reduced width of the interacting beams and the position of their "energy center." It is shown that under certain conditions there may be a collective focusing effect when all the beams are focused at a point because of the nonlinear interaction.