Rogue waves in a top-pattern and rogue waves with breathers in Ramani equation with ω-time derivative. Stability of the steady state solution

The dynamics of waves in a two-layer fluid system using a novel proportional derivative, referred to as the (\omega)-Derivative, which is introduced here, are probed. Exact solutions for the gRE-(\omega) D are derived by implementing a unified method. The results reveal many patterns of rogue waves (RWs), solitons, and complex waves with tunneling. The onset of RWs plays a crucial role in the dynamics of water waves in oceans and overseas, and have significant implications for research in fluid physics. The stability of the steady state solution (SSS) is analyzed, which is found to be unstable when the coefficient of the highest-order dispersion exceeds a critical value. We think that this work contributes new insights into the patterns of rogue waves, which may have impact on the waves dynamics.