Modulational instabilities and optical solitons due to competition of χ(2) and χ(3) nonlinearities

Modulational instabilities and optical solitons are analyzed for dispersive (or diffractive) media with competing quadratic (χ(2)) and cubic (χ(3)) nonlinearities. Such a competition between the nonlinearities is expected for the parametric interaction between the fundamental and second harmonics that originates the so-called cascaded nonlinearity. It is shown that strong interplay between the nonlinearities can occur for both large and small values of the phase mismatch and, for example, the influence of χ(2) nonlinearities may have dramatic effect on modulational instability of continuous waves and properties of optical solitons. Solitons supported by such competing nonlinearites are analyzed, and an exact analytical solution for bright solitons is presented for special relations between parameters. It is also demonstrated that even a small contribution of quadratic nonlinearity breaks degeneracy of optical solitons and also make some of them unstable.