Three-Wave Modulational Stability and Dark Solitons in a Quadratic   Nonlinear Waveguide with Grating
  We consider continuous-wave (CW) states and dark solitons (DSs) in a system of two fundamental-frequency (FF) and one second-harmonic (SH) waves in a planar waveguide with the quadratic nonlinearity, the FF components being linearly coupled by resonant reflections on the Bragg grating. We demonstrate that, in contrast with the usual situation in quadratic spatial-domain models, CW states with the phase shift between the FF and SH components are modulationally stable in a broad parameter region in this system, provided that the CW wavenumber does not belong to the system's spectral gap. Stationary fundamental DSs are found numerically, and are also constructed by means of a specially devised analytical approximation. Bound states of two and three DSs are found too. The fundamental DSs and two-solitons bound states are stable in all the cases when the CW background is stable, which is shown by dint of calculation of the corresponding eigenvalues, and verified in direct simulations. Tilted DSs are found too. They attain a maximum contrast at a finite value of the tilt, that does not depend on the phase mismatch. At a maximum value of the tilt, which grows with the mismatch, the DS merges into the CW background. Interactions between the tilted solitons are shown to be completely elastic. 
Arthur Gubeskys and Boris A. Malomed