Breather, lump and other wave profiles for the nonlinear Rosenau equation arising in physical systems.

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.