Waves in Nonlinear Lattices: Ultrashort Optical Pulses and Bose-Einstein Condensates

Y. Sivan, G. Fibich, and M. I. Weinstein
Phys. Rev. Lett. 97, 193902 – Published 8 November 2006

Abstract

The nonlinear Schrödinger equation izA(z,x,t)+x,t2A+[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 (κ1) 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.”

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  • Received 24 April 2006

DOI:https://doi.org/10.1103/PhysRevLett.97.193902

©2006 American Physical Society

Authors & Affiliations

Y. Sivan

  • School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel

G. Fibich

  • School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

M. I. Weinstein

  • Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027, USA

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Issue

Vol. 97, Iss. 19 — 10 November 2006

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