Three-wave solitons and continuous waves in media with competing quadratic
and cubic nonlinearities
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.
Min Chen (chen@math.purdue.edu)