Collision Of The Solitons And Its Dynamic Evolution In Bose-Einstein Condensates

We study the collision and evolution of solitons in quasi-one-dimensional Bose-Einstein condensates (BECs), with different phases, different potential, based on their Gross-Pitaevskii (GP) equations.Firstly, we study the single-component BECs with attractive interaction between atoms. We get its single bright soliton solution by using the inverse scattering method and this kind of solution can also be applied to a non-integrable system. By introducing two identical bright solitons, we study their collisions with the relative phase difference AΦ= 0, π/2, π, and 3π/2 in the absence of an external potential well by means of numerical simulations. After the two solitons collide, there is no change in the shap of solitons, and there is no particle radiation. The collision phenomenon is different for different phase, but the points of their collisions are the same. In the narrow potential well, we have also introduced two identical bright solitons, and have studied their collisions with the relative phase difference AΦ= 0, π. For the two cases, the two solitons collide periodically, their shapes did not change after the collision, and the collision periods are the same for different relative phase. However, they show different characteristics at the point of collisions.Secondly, we take the soliton solutions as the initial wave function to study the collision and evolution of bright-bright solitons or bright-dark solitons in quasi-one-dimensional heteronuclear two component BECs by means of numerical simulations. If there is no external potential well, the relative phase π leads to elastic collision of the bright-bright solitons, but the solitons collide with each other along with the transference of energy for the relative phase π/2. If there is external potential well, they collide periodically with the relative phase π, but with the relative phase π/ 2, the two solitons combine to one soliton after collision, with energy escaping. For the bright-dark solitons, their bright-bright solitons in the first component collide periodically with the relative phase 0 or π. The difference of them is that the former has one collision point, but the latter has two collision points. Furthermore, we have found that the bright-bright solitons are always stable and the bright-dark solitons are unstable by introducing perturbations in the initial wave functions and solving the corresponding Bogoliubov-de Gennes equations.