TY - JOUR

T1 - Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure

AU - Fibich, G.

AU - Sivan, Y.

AU - Weinstein, M. I.

N1 - Funding Information:
We thank F. Merle and S. Bar-Ad for useful discussions. We also thank the referees for many useful comments. G. Fibich and Y. Sivan were partially supported by grant No. 2000311 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. M.I. Weinstein was supported in part by a grant from the US National Science Foundation.

PY - 2006/5/1

Y1 - 2006/5/1

N2 - We consider nonlinear bound states of the nonlinear Schrödinger equation {A formula is presented} 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 = 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 ≫ 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 minima of the microstructure are unstable to general (asymmetric) perturbations but stable relative to symmetric perturbations. In the critical case p = 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 ( 1 0 -2 ) ) perturbations can destabilize the beam. Therefore, such beams are "mathematically" stable but "physically" unstable.

AB - We consider nonlinear bound states of the nonlinear Schrödinger equation {A formula is presented} 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 = 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 ≫ 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 minima of the microstructure are unstable to general (asymmetric) perturbations but stable relative to symmetric perturbations. In the critical case p = 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 ( 1 0 -2 ) ) perturbations can destabilize the beam. Therefore, such beams are "mathematically" stable but "physically" unstable.

KW - Bose-Einstein Condensation (BEC)

KW - Collapse

KW - Homogenization

KW - Instability

KW - Microstructure

KW - Nonlinear waves

KW - Periodic potential

KW - Solitary waves

UR - http://www.scopus.com/inward/record.url?scp=33748105488&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2006.03.009

DO - 10.1016/j.physd.2006.03.009

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AN - SCOPUS:33748105488

SN - 0167-2789

VL - 217

SP - 31

EP - 57

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 1

ER -