Symmetry

This paper explores the integrability of the Akbota equation with various types of solitary wave solutions. This equation belongs to a class of Heisenberg ferromagnet-type models. The model captures the dynamics of interactions between atomic magnetic moments, as governed by Heisenberg ferromagnetism. To reveal its further physical importance, we calculate more solutions with the applications of the logarithmic transformation, the M-shaped rational solution method, the periodic cross-rational solution technique, and the periodic cross-kink wave solution approach. These methods allow us to derive new analytical families of soliton solutions, highlighting the interplay of discrete and continuous symmetries that govern soliton formation and stability in Heisenberg-type systems. In contrast to earlier studies, our findings present notable advancements. These results hold potential significance for further exploration of similar frameworks in addressing nonlinear problems across applied sciences. The results highlight the intrinsic role of symmetry in the underlying nonlinear structure of the Akbota equation, where integrability and soliton formation are governed by continuous and discrete symmetry transformations. The derived solutions provide original insights into how symmetry-breaking parameters control soliton dynamics, and their novelty is verified through analytical and computational checks. The interplay between these symmetries and the magnetic spin dynamics of the Heisenberg ferromagnet demonstrates how symmetry-breaking parameters control the diversity and stability of optical solitons. Additionally, the outcomes contribute to a deeper understanding of fluid propagation and incompressible fluid behavior. The solutions obtained for the Akbota equation are original and, to the best of our knowledge, have not been previously reported. Several of these solutions are illustrated through 3-D, contour, and 2-D plots by using Mathematica software. The validity and accuracy of all solutions we present here are thoroughly verified.

This study investigates the conformable nonlinear Schrödinger equation (NLSE) with self-phase modulation (SPM) and Kudryashov’s generalized refractive index, crucial for pulse propagation in optical fibers. By applying the modified simplest equation method, we derive several novel soliton solutions and investigate their dynamic behavior within the NLSE framework enhanced with a conformable derivative. The governing conformable NLSE also exhibits symmetry patterns that support the structure and stability of the constructed soliton solutions, linking this work directly with symmetry-based analysis in nonlinear wave models. Furthermore, various graphs are presented through 2D, 3D, and contour plots. These visualizations highlight different soliton profiles, including kink-type, wave, dark, and bell-shaped solitons, showcasing the diverse dynamics achievable under this model, influenced by SPM and Kudryashov’s generalized refractive index. The influence of the conformable parameter and temporal effects on these solitons is also explored. These findings advance the understanding of nonlinear wave propagation and have critical implications for optical fiber communications, where managing pulse distortion and maintaining signal integrity are vital.

We introduce the Symmetric-$\mathbb{Z}$ (Sy-$\mathbb{Z}$) family, a unified class of symmetric discrete distributions on the integers obtained by multiplying a three-point symmetric sign variable by an independent non-negative integer-valued magnitude. This sign-magnitude construction yields interpretable, zero-centered models with tunable mass at zero and dispersion balanced across signs, making them suitable for outcomes, such as differences of counts or discretized return increments. We derive general distributional properties, including closed-form expressions for the probability mass and cumulative distribution functions, bilateral generating functions, and even moments, and show that the tail behavior is inherited from the magnitude component. A characterization by symmetry and sign–magnitude independence is established and a distinctive operational feature is proved: for independent members of the family, the sum and the difference have the same distribution. As a central example, we study the symmetric Poisson model, providing measures of skewness, kurtosis, and entropy, together with estimation via the method of moments and maximum likelihood. Simulation studies assess finite-sample performance of the estimators, and applications to datasets from finance and education show improved goodness-of-fit relative to established integer-valued competitors. Overall, the Sy-$\mathbb{Z}$ framework offers a mathematically tractable and interpretable basis for modeling symmetric integer-valued outcomes across diverse domains.

In the current digital landscape, where platforms process AI-generated content and intelligent network traffic on a large scale, it is the duty of such platforms to continuously measure the reliability, trustworthiness, and security of various data streams. Driven by this practical challenge, this research develops an effective decision-support mechanism in intelligent decision-making in big-data AI-generated content and network systems. The decision problem has considered several uncertainties, including content authenticity, processing efficiency, user trust, cybersecurity, system scalability, privacy protection, and cost of computing. The multidimensional uncertainty of AI-generated information and trends in network behavior are challenging to capture in traditional crisp and fuzzy decision-making models. To fill that gap, a new Picture Fuzzy Faire Un Choix Adequat (PF-FUCA) methodology is proposed, based on multi-perspective expert assessment and better computational aggregation to improve the accuracy of rankings, symmetry, and uncertainty treatment. A case scenario comprising fifteen different alternative intelligent decision strategies and seven evaluation criteria are examined under the evaluation of four decision-makers. The PF-FUCA model successfully prioritizes the best strategies to control AI-based content and network activities to generate a stable and realistic ranking. The comparative and sensitivity analysis show higher robustness, accuracy, and flexibility levels than the existing MCDM techniques. The results indicate that PF-FUCA is specifically beneficial in settings where a large amount of data has to flow, a high uncertainty rate exists, and the variables of decision are dynamic. The research introduces a scalable and credible methodological conception that can be used to facilitate high levels of intelligent computing applications to content governance and network optimization.