Analytical and numerical treatments for the Kaup–Newell dynamical equation

The present paper examines both analytically and numerically the Kaup–Newell equation being an important class of nonlinear Schrödinger equations with lots of applications in optical fibers. The Riccati equation method is employed for the analytical study and revealed quite a number of interesting soliton solutions including bright-singular, dark-singular, and singular soliton solutions to mention a few; while the Adomian’s improved decomposition method is adopted on the hand for the numerical investigation. More, since bright, dark and singular solitons are very important types of solitons that arising in laminar jet and nonlinear dispersive media; we considered certain hyperbolic ansatz to construct exact soliton solutions for the numerical comparative study. The presented numerical scheme thus validated the exact analytical solutions and turned out to possess a high-level of accuracy. We finally provided some comparison tables and depictions to support the reported results after inviting the classical Adomian’s method to assess the devised numerical technique. •The paper examines both analytically and numerically the Kaup–Newell equation.•The Riccati equation method is employed for the analytical study and revealed quite a number of interesting soliton solutions.•The Adomian’s improved decomposition method is adopted for the numerical investigation.•Certain important soliton solutions including bright, dark and singular solitons have been considered for the numerical comparative study.•The presented numerical scheme thus validated the exact analytical solutions and turned out to possess a high-level of accuracy.