Bound states of nonlinear Schrodinger equations with a periodic nonlinear microstructure Academic Article uri icon

abstract

  • We consider nonlinear bound states of the nonlinear Schrödinger equation i∂ z ϕ (z, x)=−∂ x 2 ϕ−(1+ m (N x))| ϕ| p− 1 ϕ, in the presence of a nonlinear periodic microstructure m (N x). 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= r beam/r ms 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≫ 1 (wide beams), a perturbation analysis for N≪ 1 (narrow beams) and numerical simulations for N= O (1). In the subcritical case p< 5, beams centered at local maxima of the microstructure are stable. Furthermore, beams centered at local...

publication date

  • May 1, 2006