Exact soliton solutions of the modified simplified Camassa–Holm and modified Benjamin–Bona–Mahony equations via the subsidiary ODE method - Zhang, Ting (Frontiers in Physics, 2026)
The main objective of this article is the analytical investigation of the simplified modified Camassa–Holm (SMCH) and the modified Benjamin–Bona–Mahony (BBM) equations. The SMCH equation plays an important role in modeling shallow-water wave dynamics, nonlinear dispersive phenomena, and the propagation of solitons in fluid mechanics. The BBM equation is frequently used to describe long surface gravity waves in nonlinear dispersive media and serves as a useful alternative to the standard Korteweg–de Vries (KdV) equation in mathematical physics. To construct exact analytical soliton solutions for these nonlinear models, the subsidiary ordinary differential equation (sub-ODE) method is employed. Through an appropriate wave transformation, the governing partial differential equations are reduced to nonlinear ordinary differential equations. Our mathematical technique yields several types of soliton wave shapes, including bright, dark, solitary, and periodic solitons. Bright solitons depict localized wave peaks, whereas dark solitons reflect intensity decreases against a continuous background. The resulting analytical solutions are represented in hyperbolic and trigonometric functions that exhibit complex nonlinear behaviors, such as periodic and singular patterns. These soliton structures exhibit the complex dynamics and stability of nonlinear waves propagating in dispersive mediums. The graphical demonstration of their propagation in three-dimensional, two-dimensional, and contour forms is presented for suitable parameter values.