Modulational Instability and Complex Dynamics of Confined Matter-Wave Solitons

We study the formation of bright solitons in a Bose-Einstein condensate of $^7$Li atoms induced by a sudden change in the sign of the scattering length from positive to negative, as reported in a recent experiment (Nature {\bf 417}, 150 (2002)). The numerical simulations are performed by using the 3D Gross-Pitaevskii equation (GPE) with a dissipative three-body term. We show that a number of bright solitons is produced and this can be interpreted in terms of the modulational instability of the time-dependent macroscopic wave function of the Bose condensate. In particular, we derive a simple formula for the number of solitons that is in good agreement with the numerical results of 3D GPE. By investigating the long time evolution of the soliton train solving the 1D GPE with three-body dissipation we find that adjacent solitons repel each other due to their phase difference. In addition, we find that during the motion of the soliton train in an axial harmonic potential the number of solitonic peaks changes in time and the density of individual peaks shows an intermittent behavior. Such a complex dynamics explains the ``missing solitons'' frequently found in the experiment.