Volume 106

This study employs Hirota’s bilinear method to derive exact solutions for the coupled discrete non-local nonlinear Schrödinger (NLS) equation. The equation under investigation is derived from the non-local reduction of the coupled discrete nonlinear NLS equation, which arises in various physical contexts such as nonlinear optics and Bose-Einstein condensates. Exact solutions of coupled discrete non-local NLS equations are obtained, including bright-bright one-soliton solutions, two-soliton solutions, and dark-dark soliton solutions. For the dark-dark soliton solution, the construction of the solution and the bilinear expansion are derived from the continuous system, but the continuous system solved in this way yields a breathing solution, however, in this coupled discrete non-local NLS equation, under specific parameters, we obtain coupled dark-dark soliton waves. In addition, periodic solutions, singular solutions and double spatial period solutions are obtained by taking different parameters. The soliton dynamics are visualized using mathematical software, providing insights into their behavior and interactions. This work enhances the understanding of soliton solutions in discrete non-local systems and provides a practical approach for analyzing similar nonlinear wave phenomena.

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This paper explores the reasons and practical implications of hypothesis testing as an important tool for decision making in today's data-driven world. Beginning with the seminal work of Ronald A. Fisher in the early 1900s, the paper traces how the concepts of the null hypothesis and the p-value evolved into a formal framework. Through applications in fields as diverse as automotive engineering and product development to financial auditing and policy analysis, the focus is on how hypothesis testing can convert data from uncertainty to certainty. The methodology is particularly effective in situations where data are limited and where tools such as t-distributions help to balance the risk of Type I and Type II errors. In addition, the paper highlights the role of hypothesis testing in promoting objective, evidence-based decision making while minimizing personal bias. Finally, it provides a forward-looking perspective on the integration of real-time analytics and increased transparency in research practices, as well as on the development and outlook for future hypothesis testing methods. Overall, the findings suggest that hypothesis testing remains indispensable for resolving uncertainty and guiding reliable decision making in scientific research and common problems.

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