Solitary waves in asymmetric coupled waveguides with quadratic nonlinearity

By means of direct numerical methods, we study spatial solitons and their stability in a pair of asymmetric linearly coupled waveguides with intrinsic quadratic nonlinearity. Two cases are considered in detail, viz., when the coupling constants at the fundamental and second harmonics are equal, and when the coupling at the second harmonic is absent. These cases correspond to the physical situations in which the coupled waveguides are, respectively, closely or widely separated. Two different kinds of the asymmetry between the waveguides are considered. The first corresponds to a difference in the phase mismatch between the fundamental and second harmonics in the two cores. Unfoldings of the previously known bifurcation diagrams for the symmetric coupler are studied in detail, and the stability of different branches of the solutions are tested. Simulations of dynamical evolution of unstable solitons demonstrate a trend of their rearrangement into stable solitons coexisting with them. The second kind of asymmetry is the special case when one waveguide is linear, while the other one possesses quadratic nonlinearity. In contrast to the case when both waveguides are nonlinear, in this case the soliton solutions for the two limiting cases of closely and widely separated waveguides are not much different. All the solitons in this system are found to be stable. The obtained results, and especially bifurcations between solitons of different types, suggest straightforward applications to all-optical switching.