Analytical approaches to the Kaup–Newell model in mathematical physics

This study investigates the Kaup–Newell equation, a nonlinear Schrödinger-type model with important applications in plasma physics and nonlinear optics, particularly in modeling sub-picosecond pulses. By employing the modified direct algebraic method and the advanced [Formula: see text]-expansion method, we derive a broad spectrum of soliton solutions, including hyperbolic, trigonometric, shock, singular, and mixed types. These solutions, validated using Maple software, enhance the understanding of nonlinear wave behavior. Additionally, the energy balance method is applied to examine traveling wave dynamics, yielding periodic solution in cosine form. Graphical representations in 2D, 3D, and contour plots are provided to visualize the solution profiles. The results not only demonstrate the efficacy of the proposed methods but also contribute novel insights into the Kaup–Newell model, with potential applicability to a wider class of nonlinear partial differential equations.