Rogue-wave structures for a generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

Abstract This study explores the behavior of higher-order rogue waves within a (3+1)-dimensional generalized nonlinear wave equation in liquid-containing gas bubbles. It creates the investigated equation’s Hirota D -operator bilinear form. We employ a generalized formula with real parameters to obtain the rogue waves up to the third order using the direct symbolic technique. The analysis reveals that the second and third-order rogue solutions produce two and three-waves, respectively. To gain deeper insights, we use the Cole-Hopf transformation on the transformed variables ξ and η to produce a bilinear equation. Using the system software Mathematica , the dynamic analysis presents the graphics for the obtained solutions in transformed ξ , η, and original spatial-temporal coordinates x , y , z , t . These visualizations reveal rogue waves’ intricate structure and evolution, capturing their localized interactions and significant presence within nonlinear systems. We demonstrate that rogue waves, characterized by their substantial height and sudden appearance, are prevalent in various nonlinear events. The equation examined in this study offers valuable insights into the evolution of longer waves with smaller amplitudes, which is particularly relevant in fields such as fluid dynamics, dispersive media, and plasmas. The implications of this research extend across multiple scientific domains, including fiber optics, oceanography, dusty plasma, and nonlinear systems, where understanding the behavior of rogue waves is crucial for both theoretical and practical applications.