Modulational instability conditions for the generation of localized structures in the context of matter waves in Bose-Einstein condensates are investigated analytically and numerically. The model is based on a modified Gross-Pitaevskii equation, which account for the energy dependence of the two-body scattering amplitude. It is shown that the modified term due to the quantum fluctuations modify significantly the modulational instability gain. Direct numerical simulations of the full modified Gross-Pitaevskii equation are performed, and it is found that the modulated plane wave evolves into a train of pulses, which is destroyed at longer times due to the effects of quantum fluctuations.

J. M. Gerton, D. Strekalov, I. Prodan and R. G. Hulet, “Direct Observation of Growth and Collapse of a Bose-Einstein Condensate with Attractive Interaction,” Nature, Vol. 408, 2000, pp. 692-695. doi:10.1038/35047030B

T. S. Raju, P. K. Panigrahi and K. Porsezian, “Modulational Instability of Two-Component Bose-Einstein Condensates in a Quasi-One Dimensional Geometry,” Physics Review A, Vol. 71, No. 3, 2005, Article ID: 035601.
doi:10.1103/PhysRevA.71.035601

A. Hasegawa and F. Tappert, “Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers,” Applied Physics Letter, Vol. 23, No. 3, 1973, pp. 142-144.
doi:10.1063/1.1654836

C. Tabi, A. Mohamadou and T. C. Kofane, “Modulated Wave Packets in DNA and Impact of Viscosity,” Chinese Physics Letter, Vol. 26, No. 6, 2009, Article ID: 068703.
doi:10.1088/0256-307X/26/6/068703

S. Cowell, H. Heiselberg, I. E. Mazets, J. Morales, V. R. Pandhari-pande and C. J. Pethick, “Cold Bose Gases with Large Scattering Lengths,” Physics Review Letter, Vol. 88, 2002, pp. 210403-210406.
doi:10.1103/PhysRevLett.88.210403

E. Tiesinga, C. J. Williams, F. H. Mies and P. S. Julienne, “Interacting Atoms under Strong Quantum Confinement,” Physics Review A, Vol. 61, No. 6, 2000, Article ID: 063416. doi:10.1103/PhysRevA.61.063416

H. Fu, Y. Wang and B. Gao, “Beyond the Fermi Pseudopotential: A Modified Gross-Pitaevskii Equation,” Physics Review A, Vol. 67, No. 5, 2003, Article ID: 053612.
doi:10.1103/PhysRevA.67.053612

G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis and V. V. Konotop, “Modulational Instability of Gross-Pitaevskii-Type Equations in 1+1 Dimensions,” Physics Review A, Vol. 67, No. 6, 2003, Article ID: 063610. doi:10.1103/PhysRevA.67.063610

P. Muruganandam and S. Adhikari, “Fortran Programs for the Time-Dependent Gross-Pitaevskii Equation in a Fully Anisotropic Trap,” Computer Physics Communication, Vol. 180, No. 10, 2009, pp. 1888-1912.
doi:10.1016/j.cpc.2009.04.015

L.Wu and J. F. Zhang, “Modulational Instability of (1+1)-Dimensional Bose-Einstein Condensate with Three-Body Interatomic Interaction,” Chinese Physics Letter, Vol. 24, No. 6, 2007, pp. 1471-1474.
doi:10.1088/0256-307X/24/6/012

E. Wamba, A. Mohamadou and T. C. Kofane, “Modulational Instability of a Trapped Bose-Einstein Condensate with Two-and Three-Body Interactions,” Physics Review E, Vol. 77, No. 4, 2008, Article ID: 046216.
doi:10.1103/PhysRevE.77.046216

J. K. Xu, “Modulational Instability of the Trapped Bose-Einstein Condensates,” Physics Letter A, Vol. 341, No. 5-6, 2005, pp. 527-531.
doi:10.1016/j.physleta.2005.05.018