In this paper, we present a fully discrete and structure-preserving scheme for the nonlinear fractional Schrödinger equations. The key is to introduce a scalar auxiliary variable and rewrite the equations as a new family of systems. The new systems are approximated by using the implicit midpoint rule, the repeated trapezoidal rule and the fractional centered difference method. It is shown the fully discrete scheme is mass- and energy-conserved. This is sharp contrast to the former result, i.e., the fully discrete scheme is only mass-conserved for the original equations.

中文翻译：

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非线性分数Schrödinger方程的质量和能量守恒差分格式

在本文中，我们为非线性分数阶Schrödinger方程提供了一种完全离散且保留结构的方案。关键是引入标量辅助变量并将方程式重写为一个新的系统族。通过使用隐式中点法则，重复梯形法则和分数中心差法来近似新系统。结果表明，完全离散的方案在质量和能量上都是节省的。这与以前的结果形成鲜明对比，即，完全离散的方案仅对于原始方程式是质量守恒的。