Nonlinear wave propagation in bubbly and dispersive dynamical system: stability and bifurcation analysis

This paper presents an extensive analytical and dynamic study of a (2+1)-dimensional, fourth-order Korteweg-de Vries (KdV4) equation arising in nonlinear wave motion and bubbly fluid dynamics. The KdV4 equation arises quite naturally in the study of gas bubble dynamics in liquids, which is a phenomenon of great interest in physics, engineering, and biomedicine, where the interaction between bubbles and liquids determines the characteristics of wave propagation. The bubbly fluid dynamics in liquids displays highly nonlinear phenomena due to the interaction between the oscillations of the bubbles, the compressibility of the liquid, and dispersion, making the KdV4 model of great interest for the study of acoustic wave propagation in bubbly fluids, cavitation dynamics, and multiphase flows. Using the extended modified auxiliary equation mapping method (EMAEMM), this paper obtains a number of newly generalized exact traveling wave solutions in an analytically explicit form. The solutions obtained describe a wide variety of nonlinear wave phenomena, including bright solitons, dark solitons, kink solitons, bright curved solitons, and periodic (cnoidal) solitons, which are obtained by choosing different parametric values. To better demonstrate their significance, the solutions are graphically represented in two and three dimensions using surface plots, density plots, contour plots, and projections. Furthermore, the KdV4 equation can be reduced into a planar dynamical system using the Galilean transformation. Moreover, the equilibrium points, such as centers, saddle nodes, and cusp nodes, can be examined qualitatively based on the qualitative behaviors of the various regions in the parameter space. In addition, the chaotic behavior and the presence of quasi-periodic motions for the perturbed systems can be examined using the phase planes and the Poincare maps so as to support the complexity of the behaviors of the systems. A Hamiltonian stability analysis can be used for the examination of the stability of the majority of the soliton solutions obtained within a specific range of the defined parameters, and the result can be explained numerically. Moreover, it has been seen that the sensitivity analysis of the obtained results shows high dependency on the initial conditions, and it can be used to explain the nonlinear and chaotic behaviors of the systems.