(untitled) (European Journal of Pure and Applied Mathematics, 2026)
This investigation delves into the coupled nonlinear Schr ̈odinger-Korteweg-de Vries (CNLS-KdV) framework, a quintessential paradigm for examining the intricate interplay between dispersive and nonlinear wave mechanics in multi-component environments. Mathematically, the CNLS-KdV configuration merges the nonlinear Schr ̈odinger formulation, governing modulation instabilities and wave dispersion phenomena, with the Korteweg-de Vries formulation, characterizing soliton genesis and wave profile steepening. From a physical standpoint, this theoretical construct proves invaluable across diverse domains, including optical pulse interactions in fiber-optic media, surface wave mechanics in shallow aquatic systems, and electrostatic wave propagation through plasma media. Via systematic implementation of the Khater III methodology alongside the enhanced Kudryashov approach, we establish precise closed-form solutions that reveal complex soliton architectures and their collisional behavior. These mathematical expressions illuminate fundamental energy transfer mechanisms and stability characteristics in coupled wave configurations, with particular emphasis on coherent soliton interactions and multi-wave patterns that have remained insufficiently characterized in existing literature. The employed theoretical methods exhibit remarkable computational effectiveness and mathematical rigor, facilitating comprehensive exploration of nonlinear evolution equations within a mathematically tractable paradigm. These discoveries strengthen theoretical comprehension of coupled nonlinear frameworks by broadening the arsenal of exact solution strategies for nonlinear partial differential equations. This advancement enriches predictive modeling approaches essential for engineering and scientific implementations. In particular, the outcomes enhance quantitative perspectives on soliton behavior in fluid mechanics, plasma physics, and nonlinear optics, with ramifications for optimizing fiber-optic communication technologies and plasma-based energy infrastructures. The research emphasizes the CNLS-KdV framework’s significance as a cornerstone model for investigating universal nonlinear wave phenomena spanning multiple disciplines.