A Computational Study of Time Dependent Nonlinear Schrödinger Equation With Cubic Nonlinearity

This research study presents a computational method to solve one-dimensional Schrödinger equation with cubic nonlinearity, which encompasses numerous important physical occurrences for instance the transmission of classical waves in nonlinear media with dispersion, nonlinear optics, water waves, etc. We use modified trigonometric cubic B-spline functions in the collocation method to discretize the equation in space variables. This approach converts the equation into a system of ordinary differential equations, which has been solved using the stability preserving R-K method. The computational complexity is observed as linear in terms of size of partition. The implementation of the developed approach is easy and the required computational work is also much less. Additionally, the solutions obtained in this approach can be located not just at the discrete mesh points \(x_{i}\) but at any location within the solution domain.