A qualitative stability analysis and dynamical solitons of the derivative nonlinear Schrödinger equation

By using both analytical and numerical methods, this paper examines the Kaup–Newell equation, a noteworthy class of derivative nonlinear Schrödinger equation family with several applications in optical fibers. Several interesting soliton solutions, including single, dark, and bright singular soliton solutions, were obtained by applying the extended modified auxiliary equation mapping methodology for the analytical investigation. This is the main way that our approach varies from other approaches that are currently in use. Numerous natural and physical sciences, such as hypothetical fluid dynamics, nonlinear fiber optics, electromagnetic attraction, computational physics, bio-mathematics, soliton movement, plasma physics, manufacturing research, quantum electronics, and radiation physics, have been profoundly impacted by recent discoveries. To provide a clearer picture of the dynamic aspects of the solutions, we have displayed the freshly discovered solutions in graphs with varying widths using Mathematica 14.0. The results were shown in a table. Additionally, as part of the qualitative evaluation of Derivative Nonlinear Schrödinger Equation, we presented Bifurcation Analysis and verified the stability of the answers we had obtained.