Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities

We study the mutual influence of quadratic and cubic nonlinearities on the propagation of the coupled fundamental and second harmonic waves in asymmetric optical media. For attractive potentials with positive coupling parameters, it is shown that, in systems with two and three transverse dimensions, mutually trapped waves can self-focus until collapse whenever their respective powers exceed some thresholds. On the contrary, coupled waves diffracting in a one-dimensional plane never collapse and may evolve towards stable solitonlike structures. For higher transverse dimension numbers, we investigate the question of forming two-component solitons and determine criteria for their stability.