Explicit Soliton and Periodic Solutions to Three-Wave System with Quadratic and Cubic Nonlinearities

Lie group theoretical method and the equation of the Jacobi elliptic function are used to study the three wave system that couples two fundamental frequency (FF) and a single second harmonic (SH) one by competing χ⁽²⁾ (quadratic) andχ⁽³⁾ (cubic) nonlinearities and birefringence. This system shares some of the nice properties of soliton system. On the phase-locked condition, we obtain large families of analytical solutions as the soliton, kink and periodic solutions of this system.