Stability of double-peaked solitons in Bragg gratings with the quadratic nonlinearity

We report systematic results for the existence and stability of double-peaked (DP) solitons in the known model including the Bragg grating, which acts on the fundamental- and second-harmonic waves, and the quadratic (chi(2)) nonlinearity, which accounts for the parametric interaction between the harmonics. We identify existence and stability regions for the DP solitons in the plane of relevant parameters (the relative Bragg reflectivity at the two harmonics, and phase mismatch q between them). We conclude that the existence region considerably expands with the soliton's velocity v, while the stability area remains nearly constant up to a critical value of v. The stability region quickly vanishes as one crosses the critical value, while the region of the existence of unstable DP solitons does not disappear. The stability is confined to negative soliton frequencies, and almost entirely to q<0. Collisions between stable moving solitons are investigated too, with a conclusion that they are always destructive.