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

Min Chen, D. J. Kaup, Boris A. Malomed

Research output: Contribution to journalArticlepeer-review


We formulate a general model of three-wave optical interactions (in the spatial domain), which combines quadratic [Formula presented] and cubic [Formula presented] nonlinearities, the latter including four-wave mixing. The model can be realized in [Formula presented] materials where an effective [Formula presented] nonlinearity is engineered by means of the quasi-phase-matching technique. Both self-focusing and self-defocusing [Formula presented] nonlinearities 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 “fast” one, in terms of the known two-component birefringent [Formula presented] model) is, chiefly, unstable, as in that model, but nevertheless a stability interval is found for it, which provides for the first example of stable fast solitons. The other FF soliton (the “slow” one, in terms of the same [Formula presented] 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 the common belief that such a soliton must always be unstable due to the parametric interaction. The 3W solitons are stable indeed if this is predicted by the VK condition, in the case when all the three components are positive. Following variation of the [Formula presented] 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 VK criterion, which is explained by consideration of the [Formula presented] 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 [Formula presented] nonlinearity, when all the solitons blow up into a turbulent state. Parallel to the solitons, continuous-wave solutions are studied too. In terms of the existence and stability, they resemble solitons of similar types.

Original languageEnglish
Pages (from-to)17
Number of pages1
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Issue number5
StatePublished - 2004


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