title: Waves in nonlinear lattices: ultrashort optical pulses and Bose-Einstein condensates.
abstract: The nonlinear Schrödinger equation i (partial differential)(z)A(z,x,t)+(inverted Delta)(2)(x,t)A+[1+m(kappax)]|A|2A=0 models the propagation of ultrashort laser pulses in a planar waveguide for which the Kerr nonlinearity varies along the transverse coordinate x, and also the evolution of 2D Bose-Einstein condensates in which the scattering length varies in one dimension. Stability of bound states depends on the value of kappa=beamwidth/lattice period. Wide (kappa>>1) and kappa=O(1) bound states centered at a maximum of m(x) are unstable, as they violate the slope condition. Bound states centered at a minimum of m(x) violate the spectral condition, resulting in a drift instability. Thus, a nonlinear lattice can only stabilize narrow bound states centered at a maximum of m(x). Even in that case, the stability region is so small that these bound states are "mathematically stable" but "physically unstable."
authors.affiliation: School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel.
authors.forename: Y
authors.initials: Y
authors.lastname: Sivan
authors.forename: G
authors.initials: G
authors.lastname: Fibich
authors.forename: M I
authors.initials: MI
authors.lastname: Weinstein