A mathematical stability analysis is presented of the two-beam interactions in quadratic optical nonlinear media that are now attracting such a lot of attention. The averaged Lagrangian is used, within a variational method, and the analysis is based upon a Gaussian trial function. The stability is governed by parameters that can be classified into two groups. One describes spatial solitonlike beam positions and propagation directions and the other describes beam sizes and phases. It is shown that the evolution of these parameters is determined by ten, coupled, ordinary differential equations. The stationary states are proved, mathematically, to be stable for all linear phase mismatch parameter values provided the perturbations are symmetric, i.e., perturbations to beam positions and directions. However, for perturbations to beam sizes or phases, it is proved that a number of stability regimes exist, together with forbidden parameter ranges. The analytical conclusions are completely borne out by computer simulations, and some typical examples are reported here.

• Received 30 June 1995

DOI:https://doi.org/10.1103/PhysRevA.52.4099