Ceesay Baboucarr, Baber Muhammad Zafarullah, Ahmed Nauman, Yasin Muhammad Waqas, Mohammed Wael W
Mathematics and Statistics Department, The University of Lahore, Lahore, Pakistan.
Mathematics Unit, The University of The Gambia, Serekunda, The Gambia.
Sci Rep. 2025 Jan 24;15(1):3067. doi: 10.1038/s41598-024-82678-z.
This work explores the mathematical technique known as the Hirota bilinear transformation to investigate different wave behaviors of the nonlinear Rosenau equation, which is fundamental in the study of wave occurrences in a variety of physical systems such as fluid dynamics, plasma physics, and materials science, where nonlinear dynamics and dispersion offer significant functions. This equation was suggested to describe the dynamic behaviour of dense discrete systems. We use Mathematica to investigate these wave patterns and obtained variety of wave behaviors, such as M-shaped waves, mixed waves, multiple wave forms, periodic lumps, periodic cross kinks, bright and dark breathers, and kinks and anti-kinks. These patterns each depict distinct qualities and behaviors of waves, offering insights into the interactions and evolution of waves. The results found that free parameters have a substantial impact on travelling waves, including their form, structure, and stability. With the aid of this software, we potray the dynamics of these waves in 3Ds, contours and densities plots, which enables us to comprehend how waves move and take on various forms. The novel component is the application of Hirota's bilinear approach to generate new form of solutions as highlighted above, analyse their interactions, and give better visualisations, which goes beyond prior soliton-focused investigations of the Rosenau problem. All things considered, our work advances our understanding of waves and nonlinear systems and demonstrates the value of mathematical techniques for understanding intricate physical phenomena. These results may have implications for a wide range of fields, including environmental science, engineering, and physics.
这项工作探索了被称为广田双线性变换的数学技术，以研究非线性罗森瑙方程的不同波动行为。该方程在流体动力学、等离子体物理学和材料科学等各种物理系统中的波现象研究中具有基础性作用，其中非线性动力学和色散发挥着重要作用。该方程被提出用于描述密集离散系统的动态行为。我们使用Mathematica来研究这些波形，并获得了多种波动行为，如M形波、混合波、多种波形、周期团块、周期交叉扭结、亮孤子和暗孤子，以及扭结和反扭结。这些模式各自描绘了波的不同特性和行为，为波的相互作用和演化提供了见解。结果发现，自由参数对行波有重大影响，包括其形式、结构和稳定性。借助该软件，我们以三维、等高线和密度图的形式描绘了这些波的动力学，这使我们能够理解波如何移动以及呈现各种形式。新颖之处在于应用广田双线性方法来生成上述新的解形式，分析它们的相互作用，并提供更好的可视化效果，这超越了以往对罗森瑙问题以孤子为重点的研究。综上所述，我们的工作推进了我们对波和非线性系统的理解，并展示了数学技术对于理解复杂物理现象的价值。这些结果可能对包括环境科学、工程和物理学在内的广泛领域产生影响。
