Solitary waves in Bragg gratings with a quadratic nonlinearity

We study the formation of solitary waves in quadratic nonlinear materials where the dispersion is provided by linear mode coupling mediated by a Bragg grating. We show that solitary wave solutions can be analytically found provided that the coupling of the second-harmonic waves considerably exceeds that of the fundamental ones. Furthermore, we numerically determine solitary wave solutions for the general case. These solutions prove to be close to the analytical ones. A nontrivial property of Bragg grating solitary waves is that they do not fill the complete parameter space where exponentially decaying functions are allowed to exist. Instead, we find internal boundaries inside this parameter space where the soliton intensify diverges. Moreover, double-hump solutions are found where a numerical propagation procedure shows that some of them are fairly robust.