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 mo dulational scales an long compared to the wavelength and period of the modulated wave. The discrete system we study is a model for second-ha rmonic generation in nonlinear optical waveguide arrays and also for e xciton waves at the interface between two crystals near Fermi resonanc e. The modulation equations predict their own breakdown by changing ty pe from hyperbolic to elliptic. Modulational stability (hyperbolicity of the modulation equations) is explicitly shown to be implied by line ar stability but not vice versa. When the plane-wave parameters vary s lowly in regions of linear stability, the modulation equations are hyp erbolic and accurately describe the macroscopic behavior of the system whose microscopic dynamics is locally given by plane waves. We show h ow the existence of Riemann invariants allows one to test modulated wa ve initial data to see whether the modulating wave will avoid all line ar instabilities and ultimately resolve into simple disturbances that satisfy the Hopf or inviscid Burgers equation. We apply our general re sults to several important limiting cases of the microscopic model in question.