Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure

We consider nonlinear bound states of the nonlinear Schrödinger equation i∂zϕ(z,x)=−∂x2ϕ−(1+m(Nx))|ϕ|p−1ϕ, in the presence of a nonlinear periodic microstructure m(Nx)m(Nx). This equation models the propagation of laser beams in a medium whose nonlinear refractive index is modulated in the transverse direction, and also arises in the study of Bose–Einstein Condensation (BEC) in a medium with a spatially dependent scattering length. In the nonlinear optics context, N=rbeam/rms denotes the ratio of beam width to microstructure characteristic scale. We study the profiles of the nonlinear bound states using a multiple scale (homogenization) expansion for N≫1N≫1 (wide beams), a perturbation analysis for N≪1N≪1 (narrow beams) and numerical simulations for N=O(1)N=O(1). In the subcritical case p<5p<5, beams centered at local maxima of the microstructure are stable. Furthermore, beams centered at local minima of the microstructure are unstable to general (asymmetric) perturbations but stable relative to symmetric perturbations. In the critical case p=5p=5, a nonlinear microstructure can only stabilize narrow beams centered at a local maximum of the microstructure, provided that the microstructure also satisfies a certain local condition. Even in this case, the stability region is very small so that small (O(10−2))(O(10−2)) perturbations can destabilize the beam. Therefore, such beams are “mathematically” stable but “physically” unstable.