In this paper, we derive an L1 Legendre–Galerkin spectral method with fast algorithm based on an efficient sum-of-exponentials (SOE) approximation for the kernel $t^{-1-\alpha }$ to solve the two-dimensional nonlinear coupled time fractional Schrödinger equations. The numerical method is stable without the Courant–Friedrichs–Lewy (CFL) conditions based on the error splitting argument technique and the discrete fractional Gronwall type inequality. At the same time, we use the fully implicit method to deal with the nonlinear terms. For the non-smooth solution, we employ the graded mesh method. Moreover, we estimate the parameters including fractional derivative index and the coefficients of the nonlinear term in the equations by applying the Cuckoo Search algorithm related to Lévy flights. Numerical examples are given to verify the theoretical analysis and confirm the effectiveness of the fast algorithm and the estimation method.

在本文中，我们基于内核的有效指数和（SOE）近似，使用快速算法推导了L1 Legendre-Galerkin谱方法 ${Ť}^{-\mathrm{1个}-\alpha }$来求解二维非线性耦合时间分数薛定ding方程。数值方法是稳定的，没有基于误差分裂参数技术和离散分数Gronwall型不等式的Courant-Friedrichs-Lewy（CFL）条件。同时，我们使用完全隐式方法来处理非线性项。对于非光滑解，我们采用渐变网格方法。此外，通过应用与Lévy航班相关的布谷鸟搜索算法，我们估计了方程中包括分数导数索引和非线性项系数的参数。数值算例验证了理论分析，验证了快速算法和估计方法的有效性。