Haar wavelet collocation method for linear and non-linear Schrödinger equation.

Abstract This paper explores the one-dimensional nonlinear time-dependent Schrödinger equation (NLSE) using the Haar wavelet approach. To approximate the derivatives in time and space, two distinct Haar series are employed. These series are used to linearize the NLSE through a straightforward iterative method, resulting in a set of linear equations. The proposed technique is structured to allow the numerical solution to be easily calculated at any arbitrary interval. The solution’s behavior, particularly the movement of single and double solitons, is clearly observable at various stages of time. The greatest errors in the real and imaginary components are computed and compared with existing literature to assess the method’s accuracy. To assess the accuracy, stability, and effectiveness of the proposed method, various test problems are presented.