The coupled nonlinear Schrödinger equation is used in simulating the propagation of the optical soliton in a birefringent fiber. Hereditary properties and memory of various materials can be depicted more precisely using the temporal fractional derivatives, and the anomalous dispersion or diffusion effects are better described by the spatial fractional derivatives. In this paper, one-step and two-step exponential time-differencing methods are proposed as time integrators to solve the space-time fractional coupled nonlinear Schrödinger equation numerically to obtain the optical soliton solutions. During this procedure, we take advantage of the global Padé approximation to evaluate the Mittag-Leffler function more efficiently. The approximation error of the Padé approximation is analyzed. A centered difference method is used for the discretization of the space-fractional derivative. Extensive numerical examples are provided to demonstrate the efficiency and effectiveness of the modified exponential time-differencing methods.

中文翻译：

---

时空分数阶耦合非线性Schrödinger方程的光学孤子解的有效指数时差方法

耦合的非线性Schrödinger方程用于模拟双折射光纤中孤子的传播。使用时间分数导数可以更精确地描述各种材料的遗传特性和记忆，并且通过空间分数导数可以更好地描述异常分散或扩散效果。本文提出一种单步和两步指数时间微分方法作为时间积分器，以数值方式求解时空分数耦合非线性Schrödinger方程，以获得光学孤子解。在此过程中，我们利用全局Padé逼近来更有效地评估Mittag-Leffler函数。分析了Padé逼近的逼近误差。中心差法用于空间分数导数的离散化。提供大量的数值示例，以证明改进的指数时差方法的效率和有效性。