Nonlinear Dynamics, Fluid Dynamics, Classical Optics, etc.-Waves in Nonlinear Lattices: Ultrashort Optical Pulses and Bose-Einstein Condensates

The nonlinear Schrödinger equation i∂zA(z,x,t)+∇2x,tA+[1+m(κx)]|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 κ=beamwidth/lattice period. Wide (κ≫1) and κ=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.”