Multi-soliton and rogue wave solutions and their interaction: Insights into nonlinear dynamics in integrable systems

Solitons are essential in order to understand integrable systems and solve nonlinear evolution equations. They maintain signal integrity in the optical fiber and represent localized waves in plasma, paying significantly to plasma behavior management in fusion research. In fiber optics, multi-soliton solutions enhance high-bandwidth signals over long distances. The interaction of solutions is important for understanding the dynamics of nonlinear wave systems, as it provides detailed insights into the solidity, activities, and development of complex wave configurations through various physical and mathematical contexts. In this article, we examine multi-soliton and rogue wave solutions of the time-dependent variable-coefficient Kadomtsev-Petviashvili (KP) equation using the Hirota direct method. We uncover novel interaction solutions between rogue waves and multi-solitons, revealing intricate wave behaviors that advance the understanding of real-world nonlinear phenomena. Furthermore, we explore the influence of free parameters on two- and three-soliton solutions through detailed 2D and 3D visualizations, providing a comprehensive perspective on their structural properties and interactions. These results contribute to the broader understanding of nonlinear wave behavior and their practical implications in optical, plasma, and fluid dynamics systems.