Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure
نویسنده
چکیده
We consider nonlinear bound states of the nonlinear Schrödinger equation i∂zφ(z, x) = −∂2 xφ− (1 + m(Nx))|φ|p−1φ, in the presence of a nonlinear periodic microstructure m(Nx). This equation models the propagation of laser beams in a medium whose nonlinear refractive index is modulated in the transverse direction, and can also arise in the study of BoseEinstein Condensation (BEC). In the nonlinear optics context, N = rbeam/rms denotes the ratio of beam width to microstructure characteristic scale. We study nonlinear bound state profiles using a multiple scale (homogenization) expansion for N À 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 a local maximum of the microstructure are stable, whereas beams centered at a local minimum of the microstructure are unstable to general (asymmetric perturbation), while conditionally stable relative to symmetric perturbations. In the critical case p = 5, a nonlinear microstructure can stabilize only 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 narrow.
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