Soliton states in wave mixing and third-harmonic generation

We examine the solitary-wave or soliton states in the nonlinear process involving waves of different frequencies interacting through the third-order nonlinear susceptibility in a uniform self-focusing medium. For the wave-mixing case (involving pump, Stokes, and anti-Stokes waves), it is found that there can be three solitary-wave states associated with each frequency shift below a critical frequency shift (Δω=Δ${\mathrm{\omega }}_{\mathit{c}1}$), one of which is stable and the other two unstable. However, above this critical frequency shift but below another critical frequency shift Δ${\mathrm{\omega }}_{\mathit{c}2}$ (${\mathrm{\omega }}_{\mathit{c}1}$) each frequency shift corresponds to one stable solitary-wave state. In the case of the third-harmonic generation, we have also found a solitary-wave solution. Unfortunately, the solitary-wave state in this case is unstable due to the presence of the complex growth rate of the linearized equations.