The modified cubic Benjamin–Ono equation describing internal solitary waves in the deep ocean and its related properties

In this article, we investigate the propagation of internal solitary waves in deep ocean. Based on the principles of nonlinear theory, perturbation expansion, and multi-scale analysis, a time-dependent modified cubic Benjamin–Ono (mCBO) equation is derived to describe internal solitary waves in the deep ocean with stronger nonlinearity. When the dispersive term ∂3f∂X3 vanishes, the mCBO equation transforms into the cubic BO equation. Similarly, when the dispersive term ∂3f∂X3 becomes zero and the nonlinear term ∂f3∂X degenerates into ∂f2∂X, the mCBO equation reduces to the BO equation. Furthermore, if the integral term ∂2∂X2ℵ(f) disappears, it simplifies to the mKdV equation. To gain deeper insight into the characteristics of solitary waves, conservation of mass and momentum associated with them are discussed. By employing Hirota's bilinear method, we obtain soliton solutions for the mCBO equation and subsequently investigate interactions between two solitary waves with different directions, leading to the occurrence of important events such as rogue waves and Mach reflections. Additionally, we explore how certain parameters influence Mach stem while drawing meaningful conclusions. Our discoveries reveal the complex dynamics of internal solitary waves within the deep ocean and contribute to a broader understanding of nonlinear wave phenomena.