Analysis of Multiform Optical Solitons Structures for a Dual-Mode Nonlinear Schrödinger Model under Square Root Kerr Law

This study investigates a novel dual-mode resonant nonlinear Schrödinger equation (RNLSE) incorporating the Bohm potential and governed by the square root Kerr law, a physically signi cant nonlinearity arising in nonlinear optics and quantum uids. Motivated by applications in ber communication, optical pulse propagation, and nonlinear engineering systems, we utilize two powerful analytical approaches: the enhanced modi ed extended tanh function method (eMETFM) and the \( \left( \frac\boldsymbol \mathcalG' \boldsymbol \mathcalG^2 \right) \) -expansion method. These techniques are utilized to obtain a variety of exact wave solutions to the governing RNLSE. The solutions obtained include hyperbolic, trigonometric, and rational forms, each capturing distinct physical regimes and wave structures relevant to dual-mode optical systems. The implementation of a suitable wave transformation reduces the RNLSE into ordinary di erential form, facilitating the application of the respective methods. To demonstrate the behavior of the resulting solitonic solutions, we present 2D plots, 3D surface visualizations, and contour projections. These graphical interpretations reveal di erent solitary wave patterns and amplitude pro les, showcasing their stability and structural dynamics. The results not only emphasize the rich wave phenomena inherent in the considered dual-mode system but also con rm the applicability and reliability of the proposed techniques for analyzing nonlinear wave equations in advanced physical and engineering contexts. The outcomes are expected to contribute meaningfully to the modeling and design of devices in laser systems, ber optics, and quantum communication channels.