Fractional nonlinear doubly dispersive equations: Insights into wave propagation and chaotic behavior

Abstract

The model is noteworthy for its nonlinearity and ability to describe the propagation of nonlinear waves inside the elastic, inhomogeneous Murnaghan’s rod equation. The fractional nonlinear doubly dispersive equation is applied in physics to model wave propagation in elastic materials and plasma, used to describe seismic wave dynamics and nonlinear effects in quantum systems. We used the β-fractional and M-truncated fractional derivatives to solve the fractional version of Murnaghan’s rod equation. The generalized Riccati equation mapping method has been utilized to find some new exact traveling wave solutions to the space–time fractional nonlinear doubly dispersive equation. It was observed that the singularity of water waves is influenced by the velocity and wave number parameters of soliton waves. To visually represent the solitons and categorize them, we utilized graphs, which revealed a diverse range of wave patterns that undergo changes corresponding to different values of σ and λ. In this study, we also added the analysis of chaotic behavior for the governed equation. By varying some parameters, the system’s transition from chaotic to quasi-periodic behavior is explored which highlights the impact of δ and φ on the systems dynamics. This study also investigates the phenomenon of multistability in a perturbed dynamical system, where multiple distinct behaviors, such as periodicity, quasi-periodicity and chaos, can coexist under the same set of parameters but different initial conditions. Through a detailed analysis of the system’s dynamics, we explored how varying initial constraints impact the evolution of the system.

Keywords

• Generalized Riccati equation mapping method (GREMM)
• Double dispersive equation (DDE)
• Murnaghan’s rod equation (MRE)
• Chaotic behavior
• Fractional derivatives
• Nonlinear equations.