Solitons in coupled waveguides with quadratic nonlinearity

We consider a model of two linearly coupled second-harmonic-generating waveguides. The analysis is focused on the case of no walkoff and full matching. We demonstrate existence of a bifurcation that transforms obvious symmetric soliton states into nontrivial asymmetric ones. The bifurcation point is found exactly, while a full analytical description of the asymmetric solutions is obtained by means of the variational approximation. Comparing this with numerical results generated by the shooting method, we conclude that, in a part of the range where the asymmetric states are predicted, the analytical approximation provides very good accuracy, while in another part, the asymmetric solitons disappear. Whenever they exist, however, direct partial differential equation simulations demonstrate that they are stable, while the symmetric ones are not. We also demonstrate that the asymmetric solitons remain stable if walkoff is added. The soliton states found here can be used for optical switching.