Macroscopic dynamics in quadratic nonlinear lattices

Fully nonlinear modulation equations are obtained for plane waves in a discrete system with quadratic nonlinearity, in the limit when the modulational scales an long compared to the wavelength and period of the modulated wave. The discrete system we study is a model for second-harmonic generation in nonlinear optical waveguide arrays and also for exciton waves at the interface between two crystals near Fermi resonance. The modulation equations predict their own breakdown by changing type from hyperbolic to elliptic. Modulational stability (hyperbolicity of the modulation equations) is explicitly shown to be implied by linear stability but not vice versa. When the plane-wave parameters vary slowly in regions of linear stability, the modulation equations are hyperbolic and accurately describe the macroscopic behavior of the system whose microscopic dynamics is locally given by plane waves. We show how the existence of Riemann invariants allows one to test modulated wave initial data to see whether the modulating wave will avoid all linear instabilities and ultimately resolve into simple disturbances that satisfy the Hopf or inviscid Burgers equation. We apply our general results to several important limiting cases of the microscopic model in question.