Capturing of solitons collisions and reflections in nonlinear Schrödinger type equations by a conservative scheme based on MOL

Abstract

In this work, we develop an efficient numerical scheme based on the method of lines (MOL) to investigate the interesting phenomenon of collisions and reflections of optical solitons. The established scheme is of second order in space and of fourth order in time with an explicit nature. We deduce stability restrictions using the von Neumann stability analysis. We consider a $(2+ 1)$ ( 2 + 1 ) -dimensional system of a coupled nonlinear Schrödinger equation as a general model of nonlinear Schrödinger-type equations. We consider several numerical experiments to demonstrate the robustness of the scheme in capturing many scenarios of collisions and reflections of the optical solitons related to nonlinear Schrödinger-type equations. We verify the scheme accuracy through computing the conserved invariants and comparing the present results with some existing ones in the literature.

Approximation properties of complex genuine α–Bernstein‐Durrmeyer operators

“…Several types of complex operators were studied by a number of authors (see literature 10–19 ). For some other applications in the related area, we refer readers to previous works 20–25 …”

“…For some other applications in the related area, we refer readers to previous works. [20][21][22][23][24][25] Motivated from the real case and overconvergence properties of above work on complex operators, we consider the complex form of the operators (1.2), for n ∈ N, 𝛼 ∈ [0, 1], z ∈ C and 𝑓 ∶ [0, 1] → C, a complex valued analytic function as follows:…”