Multiwavelet optimized finite difference method to solve nonlinear Schrödinger equation in Optical Fiber

Propagation of light through optical fiber is governed by partial differential equations (PDEs). Numerical solution to partial differential equations has drawn a lot of research interest recently. Multiwavelet based methods are among the latest techniques in such problems. Finite difference method (FDM), powered by its simplicity is considered as one among the popular methods available for the numerical solution of PDEs. But this technique fails to produce better result in problems like propagation of light pulses in a fiber medium, due to the presence of sharp variation in the intensity of light over a small section of the fiber. In such cases, to achieve a given accuracy FDM techniques require very small grid size throughout the region of interest. This results in high computational overhead. In this paper a new method-'multiwavelet optimized finite difference' (MWOFD) is proposed to overcome the drawback of FDM. In the proposed method, multiwavelets are used to adjust the grid size adaptively. Finer grids are placed in those regions where the intensity is high and a coarser grid where the intensity values are small. Once the grid size is optimized, FDM is used to obtain the solution. This method is highly converging and requires only less number of grids to achieve a given accuracy when contrasted with FDM. The method is demonstrated for nonlinear Schrodinger equation that governs the light propagation in optical fiber systems.