Structure-Preserving Exponential Schemes with High Accuracy for the 2D/3D Nonlinear Schrödinger Type Equation

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 nonlinear Schrödinger 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 approximates the exponential system in the time direction and leads to a semi-discrete conservative scheme. Subsequently, the Fourier pseudo-spectral method is applied to approximate the 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 developed schemes at last.