Structure-Preserving Exponential Schemes with High Accuracy for the 2D/3D Nonlinear Schrödinger Type Equation (Advances in Applied Mathematics and Mechanics, 2025)
The paper proposes a family of novel arbitrary high-order structure-preserving exponential schemes for the nonlinear Schrödinger equation. First, we introduce a quadratic auxiliary variable to reformulate the original nonlinearSchrödinger equation into an equivalent equation with modified energy. With that, the Lawson transformation technique is applied to the equation and deduces a conservative exponential system. Then, the symplectic Runge-Kutta method approximatesthe exponential system in the time direction and leads to a semi-discrete conservativescheme. Subsequently, the Fourier pseudo-spectral method is applied to approximatethe space of the semi-discrete to obtain a class of fully-discrete schemes. The constructed schemes are proved to inherit quadratic invariants and are stable. Some numerical examples are given to confirm the accuracy and conservation of the developedschemes at last.