The dynamics of surface wave propagation based on the Benjamin Bona Mahony equation

Modulation instability is one of the consequences of the water medium's inclination. It causes surface water waves to run into phenomena of splitting and merging in their propagation. An increase in wave amplitude follows this phenomenon, which can encourage the appearance of extreme waves. It is known that the Benjamin Bona Mahony (BBM) wave has modulation instability in its propagation, with the envelope evolving by the equation Nonlinear Schrodinger (NLS) equation dynamic. One of the NLS equation solution is known as Soliton on Finite Background (SFB). SFB is a continuation of the Benjamin-Feir nonlinear terms. Here, the probe of the BBM wave dynamics is conducted by transforming the complex amplitudes form of SFB variable into the polar form of displaced phase-amplitude. It was done to observe changes in the amplitude of the wave in a complex plane with phases that depend only on position. The description of the dynamics of the SFB can be illustrated through Argand diagrams. It was found that the modulation frequency affects the SFB phase: the smaller the modulation frequency, the higher the phase angle. Also, it is found that the phenomenon of SFB phase singularity occurs in extreme waves for certain frequency modulation intervals.