In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrödinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented by applying exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space. The proposed scheme is a linear system and can be solved efficiently. Numerical experiments are displayed to verify the conservation, efficiency, and good performance at a relatively large time step in long time computations.

在本文中，我们研究了哈密顿结构，并为二维分数阶非线性Schrödinger方程开发了一种新型的能量守恒方案。首先，我们给出分数阶Laplacian泛函的变分导数，以导出方程的Hamiltonian公式，并通过定义标量变量来获得等效系统。然后通过对空间中的时间积分和傅立叶伪谱离散化应用指数时间差近似来提出一种节能方案。所提出的方案是线性系统，可以有效地解决。显示了数值实验，以在较长时间的计算中以相对较大的时间步验证养护，效率和良好的性能。