Multiple Soliton Solutions, Soliton-Type Solutions and Hyperbolic Solutions for the Benjamin–Bona–Mahony Equation with Variable Coefficients in Rotating Fluids and One-Dimensional Transmitted Waves

Abstract With the help of symbolic computation, the Benjamin–Bona–Mahony (BBM) equation with variable coefficients is presented, which was proposed for the first time by Benjamin as the regularized long-wave equation and originally derived as approximation for surface water waves in a uniform channel. By employing the improved (G′/G)$(G^' /G)$-expansion method, the truncated Painlevé expansion method, we derive new auto-Bäcklund transformation, hyperbolic solutions, a variety of traveling wave solutions, soliton-type solutions and two solitary wave solutions of the BBM equation. These obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves.