Based on the invariant energy quadratization approach, we propose a linear implicit and local energy preserving scheme for the nonlinear Schrödinger equation with wave operator, that describes the solitary waves in physics. In order to overcome the difficulty of designing an efficient scheme for the imaginary functions of the nonlinear Schrödinger equation with wave operator, we transform the original problem into its real form. By introducing some auxiliary variables, the real form of nonlinear Schrödinger equation with wave operator is reformulated into an equivalent system, which admits the modified local energy conservation law. Then the equivalent system is discretized by the finite difference method to yield a linear system at each time step, which can be efficiently solved. A numerical analysis of the proposed scheme is conducted to show its uniquely solvability and convergence. Our proposed method is validated by numerical simulations in terms of accuracy, energy conservation law and stability.

基于不变能量正交化方法，我们为非线性薛定wave方程提出了一种线性隐式和局部能量守恒方案，该非线性薛定ö方程具有波动算子，描述了物理学中的孤立波。为了克服使用波动算子为非线性Schrödinger方程的虚函数设计有效方案的困难，我们将原始问题转化为它的实数形式。通过引入一些辅助变量，将带有波动算子的非线性薛定ding方程的实形式重新构造为等效系统，从而接受了修改后的局部能量守恒定律。然后，通过有限差分法将等效系统离散化，从而在每个时间步长生成线性系统，可以有效地求解该线性系统。对提出的方案进行了数值分析，以显示其独特的可解性和收敛性。通过数值模拟验证了我们提出的方法的准确性，节能规律和稳定性。