Solitary Wave Solutions and Modulation Instability of First- and Second-Order Benjamin–Ono Fluid Dynamics Models With Applications

The first- and second-order wave equations of Benjamin–Ono control the propagation of nonlinear Rossby waves in a rotating fluid and define a broad class of internal waves in a stratified fluid of enormous depth. New solitary wave solutions and the modulation instability of the solutions for the first- and second-order Benjamin–Ono partial differential equations are studied in this paper. Using the extended simple equation approach, we obtained new exact, periodic, kink soliton, dark soliton, and bright soliton solutions. The moments of the obtained solutions are shown graphically to illustrate the models’ physical properties. These solutions have important uses in nanofluids and fluid dynamics. Using modulation instability analysis, the sensitivities of the septic nonlinearities in the models are examined. All of the resulting solutions are analytical and stable under the specified conditions, according to the modulation instability analysis.