Bifurcation and chaos: Unraveling soliton solutions in a couple fractional-order nonlinear evolution equation

Abstract

Shallow water waves represent a significant and extensively employed wave type in coastal regions. The unconventional bidirectional transmission of extended waves across shallow water is elucidated through nonlinear fractional partial differential equations, specifically the space–time fractional-coupled Whitham–Broer–Kaup equation. The application of two distinct analytical methods, namely, the generalized logistic equation approach and unified approach, is employed to construct various solutions such as bright solitons, singular solitary waves, kink solitons, and dark solitons for the proposed equation. The physical behavior of calculated results is graphically represented through density, two- and three-dimensional plots. The obtained solutions could have significant implications across a range of fields including plasma physics, biology, quantum computing, fluid dynamics, optics, communication technology, hydrodynamics, environmental sciences, and ocean engineering. Furthermore, the qualitative assessment of the unperturbed planar system is conducted through the utilization of bifurcation theory. Subsequently, the model undergoes the introduction of an outward force with the aim of inducing disruption, resulting in the emergence of a perturbed dynamical system. The detection of chaotic trajectory in the perturbed system is accomplished through the utilization of a variety of tools designed for chaos detection. The execution of the Runge–Kutta method is employed to assess the sensitivity of the examined model. The results obtained serve to underscore the effectiveness and applicability of the proposed methodologies for the assessment of soliton structures within a broad spectrum of nonlinear models.

Keywords

• coupled fractional whitham–broer–kaup model
• the generalized logistic equation method
• unified method
• soliton structures
• phenomena of bifurcation and chaos
• sensitivity