Mismatch management for optical and matter-wave quadratic solitons
  We propose a way to control solitons in $\chi ^{(2)}$ (quadratically-nonlinear) systems by means of periodic modulation imposed on the phase-mismatch parameter ("mismatch management", MM). It may be realized in the co-transmission of fundamental-frequency (FF) and second-harmonic (SH) waves in a planar optical waveguide via a long-period modulation of the usual quasi-phase-matching pattern of ferroelectric domains. The MM may also be implemented by dint of the Feshbach resonance in a harmonically-modulated magnetic field in a hybrid atomic-molecular Bose-Einstein condensate (BEC), with the atomic and molecular mean fields (MFs) playing the roles of the FF and SH, respectively. The problem is analyzed by two methods. First, we identify stability regions for spatial solitons in the MM system, in terms of the MM amplitude and period, using the MF equations for spatially-inhomogeneous configurations. In particular, an instability enclave is found inside the stability area.The robustness of the solitons is also tested against variation of the shape of the input pulse, and a threshold for the formation of stable solitons is found in terms of its power. Interactions between solitons are virtually unaffected by the MM. The second method (\textit{parametric approximation}), going beyond the MF description, is developed for spatially-homogeneous states. It demonstrates that the MF description is valid for large modulation periods, while at smaller periods the non-MF component acquires gain, which implies destruction of MF under the action of the high-frequency MM. 
R. Driben, Y. Oz, B. A. Malomed, A. Gubeskys, and V. A. Yurovsky (Tel   Aviv University)