Novel solitary wave solutions in dual-mode simplified modified Camassa-Holm equation in shallow water waves

This study addresses dual-mode equations, focusing on the nonlinear wave behavior characterized by simultaneous bidirectional movement influenced by a specific phase velocity. Building upon Korsunsky’s enhanced Korteweg-De Vries equation, we introduce an innovative dual-mode simplified modified Camassa-Holm model and systematically analyze its solutions. Employing explicit \(\tanh {/}\coth\) $tanh/coth$ , Exp-function expansion, extended Kudryashov-expansion, and rational \(\sin {/}\cos\) $sin/cos$ techniques, we establish soliton solution constraints. The validation process includes generating supportive 3D and 2D plots. Furthermore, the study is exploring the interaction between dual waves and how the speed of waves changes. The outcomes of this research are expected to unravel soliton behavior. particularly extend into diverse domains, such as shallow water waves and optical fiber communication. The novel techniques employed in this study contribute to the stability of the exact solutions obtained, offering new perspectives beyond previous endeavors in the field.