A reliable multi-resolution collocation algorithm for nonlinear Schrödinger equation with wave operator

The solution of a nonlinear hyperbolic Schrödinger equation (NHSE) is proposed in this paper using the Haar wavelet collocation technique (HWCM). The central difference technique is applied to handle the temporal derivative in the NHSE and the finite Haar functions are introduced to approximate the space derivatives. After linearizing the NHSE, it is transformed into full algebraic form with the help of finite difference and Haar wavelets approximation. Solving this well-conditional system of the algebraic equation, we obtained the required solution. Theoretical convergence and stability analysis of HWCM is also performed for two-dimensional NHSE which is supported by the experimental rate of convergence. The numerical findings for the moving soliton wave in the form of $ |\varphi | $ are explored in depth using Haar wavelets. The propagation of soliton waves is captured accordingly and the time blow-up phenomenon has also been handled by the proposed HWCM because of the well-conditional behaviour of the transformed algebraic equations. Several examples are presented to demonstrate the proposed method and the results are found correct and efficient. In the last example, we have considered a practical case that has no exact solution.