In this paper, we propose a family of high-order conservative schemes based on the exponential integrators technique and the symplectic Runge-Kutta method for solving the nonlinear Gross-Pitaevskii equation. By introducing generalized scalar auxiliary variable, the equation is equivalent to a new system with both mass and modified energy conservation laws. Then, a conservative semi-discrete exponential scheme is given by combining the symplectic Runge-Kutta method and the Lawson method in the time direction. Subsequently, we apply the Fourier pseudo-spectral method to approximate semi-discrete system in space and obtain the fully-discrete schemes that conserve the energy and mass. Numerical examples are presented to confirm the accuracy and conservation of the developed schemes.

在本文中，我们提出了一系列基于指数积分器技术和辛 Runge-Kutta 方法的高阶保守方案，用于求解非线性 Gross-Pitaevskii 方程。通过引入广义标量辅助变量，该方程等效于一个具有质量守恒律和修正能量守恒定律的新系统。然后，结合辛Runge-Kutta方法和Lawson方法在时间方向上给出了一个保守的半离散指数格式。随后，我们应用傅里叶伪谱方法来逼近空间中的半离散系统，得到能量和质量守恒的全离散方案。给出了数值例子来证实所开发方案的准确性和守恒性。