Solitary-waves of the Nonpolynomial Schrodinger Equation: Bright Solitons in Bose-Einstein Condensates

摘要

Recently we have derived an effective one-dimensional nonpolynomialSchr\"odinger equation (NPSE) [Phys. Rev. A {\bf 65}, 043614 (2002)] thataccurately describes atomic Bose-Einstein condensates under transverse harmonicconfinement. In this paper we analyze the stability of the solitary-wavesolutions of NPSE by means of the Vakhitov-Kolokolov criterion. Stableself-focusing solutions of NPSE are precisely the 3D bright solitonsexperimentally found in Bose-Einstein condensates [Nature {\bf 417}, 150(2002); Science {\bf 296}, 1290 (2002)]. These self-focusing solutionsgeneralize the ones obtained with the standard nonlinear Schr\"odinger equation(NLSE) and take into account the formation of instabilities in the transversedirection. We prove that neither cubic nor cubic-quintic approximations of NPSEare able to reproduce the unstable branch of the solitary-wave solutions.Moreover, we show that the analitically found critical point of NPSE is inremarkable good agreement with recent numerical computations of 3D GPE. We alsodiscuss the formation of multiple solitons by a sudden change of the sign ofthe nonlinear strength. This formation can be interpreted in terms of themodulational instability of the time-dependent macroscopic wave function of theBose condensate. In particular, we derive an accurate analytical formula forthe number of multiple bright solitons.