Phase plane bifurcation analysis of water wave dynamics in the simplified modified Camassa–Holm model with friction and wind effects

Abstract

The simplified modified Camassa–Holm equation plays a pivotal role in modeling nonlinear wave dynamics across diverse fields, including optical fibers, biological transport, plasma physics, and shallow water flows. Its unique mathematical structure captures essential features of wave-breaking phenomena, peakon interactions, and dispersive effects that are crucial for understanding real-world wave behavior. Motivated by the need to predict extreme wave events and design efficient wave energy systems, this study investigates how external forces such as friction and wind influence wave dynamics. We explore rich dynamical transitions through a detailed bifurcation analysis. Our systematic investigation reveals critical thresholds in parameter space where small changes in forcing conditions lead to dramatic transformations in wave behavior. We identify key equilibrium states, nodes, foci, centres, and saddle points, that govern the system’s response, leading to the discovery of novel wave solutions, including kink-like waves, periodic structures, and breather-like solitons. These soliton shapes have potential applications in coastal protection, energy harvesting from waves, and signal modulation in nonlinear optical systems, highlighting their practical significance. These solutions are rigorously validated through numerical simulations and stability analysis, confirming their physical relevance across different parameter regimes. The solutions are derived in exact analytical forms using hyperbolic and trigonometric functions, revealing how parameter variations trigger qualitative shifts in wave patterns. Specifically, we demonstrate how the wind parameter $\alpha$ controls wave amplification while the friction parameter $\beta$ governs energy dissipation, providing a complete picture of their competing effects on wave evolution. Our findings deepen the theoretical understanding of nonlinear waves while offering practical insights for coastal engineering, climate modeling, signal transmission, and wave energy systems. By explicitly linking solution families to potential engineering applications, this study provides a framework for designing devices that exploit specific soliton structures to achieve targeted wave control and energy efficiency. The methodology developed here can be readily extended to other nonlinear dispersive systems, opening new avenues for investigating wave-structure interactions in various physical contexts.