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We consider the model of a dual-core spatial-domain coupler with chi^(2) andchi^(3) nonlinearities acting in two parallel cores. We construct families ofsymmetric and asymmetric solitons in the system with self-defocusing chi^(3)terms, and test their stability. The transition from symmetric to asymmetricsoliton branches, and back to the symmetric ones proceeds via a bifurcationloop. A pair of stable asymmetric branches emerge from the symmetric family viaa supercritical bifurcation; eventually, the asymmetric branches merge backinto the symmetric one through a reverse bifurcation. The existence of the loopis explained by means of an extended version of the cascading approximation forthe chi^(2) interaction, which takes into regard the XPM part of the chi(3)interaction. When the inter-core coupling is weak, the bifurcation loopfeatures a concave shape, with the asymmetric branches losing their stabilityat the turning points. In addition to the two-color solitons, which are builtof the fundamental-frequency (FF) and second-harmonic (SH) components, in thecase of the self-focusing chi^(3) nonlinearity we also consider single-colorsolitons, which contain only the SH component, but may be subject to theinstability against FF perturbations. Asymmetric single-color solitons arealways unstable, whereas the symmetric ones are stable, provided that they donot coexist with two-color counterparts. Collisions between tilted solitons arestudied too.

Symmetric and asymmetric solitons in dual-core couplers with competing quadratic and cubic nonlinearities