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 uni...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 parameter𝛼controls wave amplification while the friction parameter𝛽governs 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.展开更多
The simplified modified Camassa-Holm (SMCH) equation is an important nonlinear model equation for identifying various wave phenomena in ocean engineering and science. The new auxiliary equation (NAE) method has been a...The simplified modified Camassa-Holm (SMCH) equation is an important nonlinear model equation for identifying various wave phenomena in ocean engineering and science. The new auxiliary equation (NAE) method has been applied to the SMCH equation. Base on the method, we have obtained some novel an- alytical solutions such as hyperbolic, trigonometric, exponential, and rational function solutions of the SMCH equation. For appropriate values of parameters, three dimensional (3D) and two dimensional (2D) graphs are designed by Mathematica. The stability of the model is also discussed in this manuscript. The dynamic and physical behaviors of the solutions derived from the SMCH equation have been ex- tensively discussed by these plots. All our solutions are indispensable for understanding the nonlinear phenomena of dispersive waves that are important in ocean engineering and science. In addition, our results are essential to clarify the various oceanographic applications containing ocean gravity waves, offshore rig in water, energy associated with a moving ocean wave and numerous other related phenom- ena. Finally, the obtained solutions are helpful for studying wave interactions in many new structures and high-dimensional models.展开更多
文摘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 parameter𝛼controls wave amplification while the friction parameter𝛽governs 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.
文摘The simplified modified Camassa-Holm (SMCH) equation is an important nonlinear model equation for identifying various wave phenomena in ocean engineering and science. The new auxiliary equation (NAE) method has been applied to the SMCH equation. Base on the method, we have obtained some novel an- alytical solutions such as hyperbolic, trigonometric, exponential, and rational function solutions of the SMCH equation. For appropriate values of parameters, three dimensional (3D) and two dimensional (2D) graphs are designed by Mathematica. The stability of the model is also discussed in this manuscript. The dynamic and physical behaviors of the solutions derived from the SMCH equation have been ex- tensively discussed by these plots. All our solutions are indispensable for understanding the nonlinear phenomena of dispersive waves that are important in ocean engineering and science. In addition, our results are essential to clarify the various oceanographic applications containing ocean gravity waves, offshore rig in water, energy associated with a moving ocean wave and numerous other related phenom- ena. Finally, the obtained solutions are helpful for studying wave interactions in many new structures and high-dimensional models.
