Numerical Solutions of the (2+1)-Dimensional Nonlinear and Linear Time-Dependent Schrödinger Equations Using Three Efficient Approximate Schemes

Abstract

In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, namely the split-step Fourier transform (SSFT), Fourier pseudo-spectral method (FPSM), and the hopscotch method (HSM). A bright 1-soliton solution is considered for the 2D NLSE, whereas a Gaussian wave solution is determined for the 2D TDSE. Although the analytical solutions of these partial differential equations can sometimes be reached, they are either limited to a specific set of initial conditions or even perplexing to find. Therefore, our suggested approximate solutions are of tremendous significance, not only for our proposed equations, but also to apply to other equations. Finally, systematic comparisons of the three suggested approaches are conducted to corroborate the accuracy and reliability of these numerical techniques. In addition, each scheme’s error and convergence analysis is numerically exhibited. Based on the MATLAB findings, the novelty of this work is that the SSFT has proven to be an invaluable tool for the presented 2D simulations from the speed, accuracy, and convergence perspectives, especially when compared to the other suggested schemes.