Waves in nonlinear lattices: Ultrashort optical pulses and bose-einstein condensates

The nonlinear Schrodinger equation i partial derivative(z)A(z,x,t) + del(2)(x,t)A + [1+m(kappa x)]vertical bar A vertical bar(2)A = 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 = 0(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."