This study investigates exact solitary wave solutions of the modified Camassa–Holm (mCH) and modified Kuramoto–Sivashinsky (mKS) equations, which are fundamental in fluid dynamics, nonlinear optics, and quantum mechanics. Solutions are obtained using the Formula: see text-expansion neural network analytical method, a hybrid approach that integrates the symbolic capability of neural networks (NNs) with Formula: see text-expansion method, enabling the direct construction of analytical exact solutions without relying on classical transformations. The method yields a rich variety of soliton structures in trigonometric, hyperbolic, and rational forms, including periodic, bright, dark, V-shaped, kink, and singular kink solitons. The dynamics are illustrated through 2D, 3D with density surface, and polar plots, providing clear physical insights. The results demonstrate the efficiency and versatility of the method in capturing complex nonlinear behaviors and suggest potential applications in nonlinear wave propagation in fluid dynamics, plasma physics, optical communications, and biological systems.
Tipu et al. (Wed,) studied this question.