BRIGHT AND DARK GAP SOLITONS GOVERNED BY QUADRATIC NONLINEARITIES

Abstract

It is shown that localized nonlinear modes associated with a gap in the frequency spectrum of linear waves, i.e. the so-called gap solitons, can be supported by purely quadratic nonlinearities. Using the asymptotic expansion technique, a system of coupled nonlinear equations describing interaction between two conter-propagating waves of the same frequency is derived and its spatially localized solutions are found in an explicit form. In some cases the equations and their localized solutions are similar to those known for cubic nonlinearities [see e.g. Yu. S. Kivshar and N. Flytzanis, Phys. Rev.A46, 7972 (1992)] but in other cases they differ very much, being described by a novel type of the couple-mode equations. Particular examples of the physical systems considered include localized gap modes in a diatomic chain on a nonlinear substrate (on-site) potential, a diatomic lattice with anharmonic interaction between nearest-neighbor particles, and spatial gap solitons due to cascaded second-order processes in an optical medium with χ(2) nonlinear susceptibility in the presence of grating. Alternative approaches for deriving the couple-mode equations of the theory of gap solitons as well as parametric stabilization of gap solitons in a lossy medium are also discussed.