Motion of nonlinear Schroedinger solitons under external potentials

自从1995年在碱金属气体中发现玻色-爱因斯坦凝聚体以来，BECS和原子激光的材料性质得到了广泛的研究。对BECS的非线性动力学也进行了深入的研究。旋转的BEC中存在量子涡旋和涡旋晶格。在外加磁场的约束下，一维空间中的BEC中存在孤子和暗孤子。这些非线性激发可以用Gross-Pitaevskii方程或非线性薛定谔方程来很好地描述。我们数值研究了外势作用下的非线性薛定谔方程和空间调制的非线性薛定谔方程，得到了一些关于孤子和涡旋的新结果。我们已经在《物理评论》等杂志上发表了这一结果。通过几束激光的干涉，我们可以引入空间周期势，这被称为光学晶格。在不同的光学晶格中，孤子和涡旋表现出不同的运动。主要研究结果如下：1.给出了一些结论。研究了正方形光学晶格中孤子晶格和涡旋晶格的稳定性。我们证明了禁带孤子稳定地存在于准周期光学晶格中。研究了旋转光格子中孤子的稳定性。研究了具有空间调制非线性的一维非线性薛定谔方程中的孤子运动。在具有空间调制非线性的非线性薛定谔方程中，我们发现了一个稳定的二维孤子。我们发现了一些自然产生二维暗孤子的过程。在无粘性的复Ginzburg-Landau方程中发现了以任意速度运动的孤子和涡旋，并研究了它们之间的相互碰撞。

Since Bose-Einstein condensates were found in alkali metal gases in 1995, material properties of BECs and atomic laser have been intensively studied. Nonlinear dynamics in BECs is also intensively studied. Quantum vortices and vortex lattices were found in rotating BECs. Solitons and dark solitons were found in BECs confined in one dimensional space by external magnetic field. These nonlinear excitation is well described by the Gross-Pitaevskii equation or the nonlinear Schroedinger equation.We have numerically studied the nonlinear Schroedinger equation under external potential and the spatially modulated nonlinear Schroedingher equation, and found some new results concerning the solitons and the vortices. We have published the results in Physical Review and so on. By the interference of several laser lights, we can introduce spatially periodic potentials, which are called optical lattices. In various optical lattices, solitons and vortices exhibit various motion. Some results are pointed out in the followings.1. We have studied the stability of soliton lattices and vortex lattices in a square optical lattice.2. We have shown that gap solitons exist stably in a quasiperiodic optical lattice.3. We have studied the stability of solitons in a rotating optical lattice.4. We have studied soliton motions in the one-dimensional nonlinear Schroedinger equation with spatially modulated nonlinearity.5. We have found a stable two-dimensional soliton in the nonlinear Schroedinger equation with spatially moduated nonlinearity.6. We have found some processes in which two-dimensional dark solitons are naturally created.7. We have found moving solitons and vortices with arbitrary velocities in the complex Ginzburg-Landau equation without viscosity, and studied the mutual collision.