非Kerr光纤中亮孤子的稳定性与相互作用

Stability and interaction of bright solitons in non-Kerr fiber

摘要:

非Kerr光纤中的亮孤子的演化可以用具有三次-五次竞争非线性项的非线性薛定谔方程来描述. 为数值求解该方程的初值问题，本文首先将无界区域截断为有界区域，根据亮孤子在远场的渐近行为构造了合理的边界条件，从而将初值问题转换为初边值问题. 对这个初边值问题，本文分别提出了Crank-Nicolson有限差分(Crank-Nicolson Finite Differene, CNFD)和时间分裂有限差分 (Time-Spliting Finite Difference, TSFD)格式. 这两种格式在空间和时间维度上都具有二阶精度，其中CNFD 格式是全隐式格式，可以守恒离散能量和质量，TSFD是线性隐式格式，可以守恒离散质量. 在以数值算例验证两种方法的计算效率后，本文用TSFD格式研究了非Kerr光纤中亮孤子的稳定性与相互作用.

Abstract:

Dynamical behavior of the bright solitons in non-Kerr fiber can be described by the nonlinear Schrodinger equation (NLSE) with cubic-quintic competing nonlinear terms. In this paper, to numerically solve the initial value problem of the NLSE, two difference schemes are proposed. Firstly, we transfer the initial value problem into the initial value problem with boundary conditions, truncate the unbounded region into a bounded region and construct a reasonable boundary condition based on the asymptotic behavior of bright solitons in the far field. Then we design the Crank-Nicolson finite difference (CNFD) and time-splitting finite difference (TSFD) schemes. The CNFD scheme is fully implicit and can conserve the discrete energy and mass. Meanwhile, the TSFD is linear implicit and can only conserve the discrete mass. Finally, after the performance of the two schemes is compared by some examples, we explore the stability and interaction of the bright solitons in non-Kerr fiber by using the TSFD scheme.