Abstract: We study the traveling wave and other solutions of the perturbed Kaup-Newell Schrödinger dynamical equation that signifies long waves parallel to the magnetic field. The wave solutions such as bright-dark (solitons), solitary waves, periodic and other wave solutions of the perturbed Kaup-Newell Schrödinger equation in mathematical physics are achieved by utilizing two mathematical techniques, namely, the extended F-expansion technique and the proposed exp\((-\phi(\xi))\)-expansion technique. This dynamical model describes propagation of pluses in optical fibers and can be observed as a special case of the generalized higher order nonlinear Schrödinger equation. In engineering and applied physics, these wave results have key applications. Graphically, the structures of some solutions are presented by giving specific values to parameters. By using modulation instability analysis, the stability of the model is tested, which shows that the model is stable and the solutions are exact. These techniques can be fruitfully employed to further sculpt models that arise in mathematical physics.

Key words: extended F-expansion method, generalized $\exp(-\phi(\xi))$-expansion technique, perturbed Kaup-Newell Schrödinger equation, traveling wave solutions