In this article, we explored the commutativity of a ring $ \Lambda $ that is equipped with a unique class of mappings called centrally extended $ n $-homoderivations, where $ n $ is an integer. These mappings generalize the concepts of derivations and homoderivations. Furthermore, we investigated specific properties of the center of such rings.

This paper aims to establish the following: Let $ \Omega $ be a ring that satisfies some conditions and has an idempotent element $ f\neq 0, 1 $. We intend to show that if $ G $ is any suitable multiplicative generalized CE-derivation of $ \Omega $, then $ G $ is a centrally extended additive.

We utilize a cohesive methodology to obtain some new solitary wave solutions for the (2 + 1)-dimensional nonlinear Schrödinger equation (2D-NLSE). The solutions provided herein are significant for elucidating physical phenomena in various domains, including optical fibers, plasma media, and ocean waves. Furthermore, scientific computing would be used to illustrate the physical interpretation of nonlinear waves. Our study examines how 2D-NLSE wave solutions affect physical model characteristics such as group velocity dispersion, nonlinearity, and linear coefficients. These variables functioned to control the amplitude and wave phase of the optical solitary waves during transmission. Finally, the strategy provided here is applicable to many nonlinear systems and new energy trends in natural science.

In this article, we examine the commutativity of a ring $\Omega$ endowed with a specific kind of mappings called centrally extended $(\Phi, m)$-homoderivations, where $\Phi$ is a mapping on $\Omega$, and $m$ is an integer. This mapping is a comprehensive kind of the homoderivation, $\Phi- $homoderivation, and $ m $-homoderivation. Besides, we provide some properties of its center.

We introduce the concept of reverse ( l , m , n )‐homoderivations as a new class of additive mappings on rings, providing a common generalization of reverse derivations, antihomomorphisms, and related homoderivation‐type mappings. After establishing their existence via suitable examples, we examine their influence on the prime and semiprime ring structures. It is shown that the action of these mappings on nontrivial ideals leads to strong structural consequences, including commutativity in prime rings and the emergence of nontrivial central ideals in semiprime rings under appropriate conditions. This approach unifies and refines several earlier results, highlighting the effectiveness of reverse ( l , m , n )‐homoderivations in capturing key commutativity and centrality phenomena.