Three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities

Abstract

We formulate a general model of three-wave optical interactions (in the spatial domain), which combines quadratic ($\chi ^{(2)}$) and cubic ($\chi ^{(3)}$) nonlinearities, the latter including four-wave mixing. The cases of both the self-focusing and self-defocusing $\chi ^{(3)}$ nonlinearity are considered. The birefringence of the two fundamental-frequency (FF) waves is taken into regard. Several types of solitons in this system are found, by means of the variational approximation and numerical methods. These are exact single-component solitons and generic three-wave (3W) ones, which are classified by relative signs of their components. Stability of the solitons is investigated by means of the Vakhitov-Kolokolov (VK) criterion, and then tested by direct simulations. One type of the single-component FF solitons (the \textquotedblleft fast\textquotedblright\ one, in terms of the known two-component birefringent $\chi ^{(3)}$ model) is, chiefly, unstable, as in that model, but nevertheless a stability interval is found for it. The other FF soliton (the \textquotedblleft slow soliton\textquotedblright , in terms of the same $\chi ^{(3)}$ model, where it is always stable) has its stability and instability regions. A single-component soliton in the second harmonic (SH) is found too; it also has its stability region, contrary to an expectation that it must always be unstable. The 3W solitons are stable indeed if this is predicted by the VK condition, in the case when all the three components are positive. With the variation of the $\chi ^{(2)}$ mismatch parameter, the 3W soliton bifurcates from the SH one, and at another point it bifurcates back into the slow-FF single-component soliton; conjectured normal forms of the respective bifurcations are given. 3W solitons with different signs of their components may be unstable contrary to the prediction of the VK criterion, which is explained by consideration of the $\chi ^{(2)}$ term in the system's Hamiltonian. In direct simulations, unstable solitons evolve into stable breathers. A different instability takes place in the case of the self-defocusing $\chi ^{(3)}$ nonlinearity, when all the solitons blow up. Parallel to the solitons, continuous-wave (CW) solutions are studied too. In terms of the existence and stability, they resemble solitons of similar types.