<p>In this paper, we study a detailed analysis of the modified Benjamin-Bona-Mahony (BBM) equation, a nonlinear partial differential equation of significant interest in the investigation of nonlinear wave propagation and optical illusion modeling, wherein wave interference and chaotic interactions create intricate intensity distributions that are similar to optical patterns found in nonlinear optical systems. This equation, which generalizes traditional wave equations to include higher-order nonlinear and dispersive effects, is an effective tool for modeling intricate wave phenomena in a variety of physical systems. The collective impact of dispersive and nonlinear effects–where nonlinearity is amplitude-dependent self-interaction and phase modulation, and dispersion is for wave spreading and velocity dispersion. We start by attempting to test the integrability of the perturbed BBM equation through the Painlevé test. This technique, famous for proving the existence of movable singularities in solutions, validates the integrability characteristics of the equation and offers interesting information regarding the structure and nature of its solutions. Successful application of the Painlevé test proves that the equation has analytic, stable solutions under particular conditions. Next, we utilize the generalized exponential rational function approach to obtain a rich variety of exact wave solutions. This method, an extension of conventional solution methods, allows for the retrieval of multiple functional forms, such as solitary wave solutions, periodic wave solutions, and rational function solutions. Every solution uncovers unique physical properties of the inherent wave dynamics, leading to an enhanced understanding of the nonlinear phenomena described by the equation. To visualize and describe these solutions, we apply a variety of graphical methods. The solutions are displayed using 2D plots to show waveform profiles, 3D surface plots to expose amplitude change in space and time, revolution plots to show symmetry and geometrical structure, and density plots to record localized intensity variation. These visual representations offer an intuitive and complete view of the dynamic behavior produced by the solutions. Besides deriving and plotting solutions, we examine their stability in the context of a Hamiltonian system. Based on an analysis of the energy conservation and perturbation response of the system, we evaluate the resilience and long-term behavior of the solutions under small perturbations, providing information regarding their physical feasibility and engineering implications. We also examine the intricate and chaotic dynamics embedded in the system. With careful numerical computations, we point out phenomena like multistability, bifurcations, and initial value sensitivity. The appearance of chaotic dynamics is illustrated with a set of diagnostic methods, such as 2D and 3D phase portraits, time-series graphs, and Poincaré maps. These diagnostics illustrate the complex pathways in phase space and the uncertain but deterministic behavior of the system under different parameters and initial conditions. Overall, this research not only improves the theoretical framework of the modified BBM equation but also gives a solid set of analytical and computational tools to investigate nonlinear wave dynamics in applied settings. The originality of the work stems from the initial implementation of the generalized exponential rational function method (GERFM) on the mBBM equation to generate a variety of analytical solutions with detailed stability and chaos analysis going beyond conventional methods. These results show dynamically stable (Lyapunov-stable) and physically realistic solutions, obtained by choosing parameter ranges that guarantee bounded, nonsingular wave behavior and rule out complex or imaginary solutions.</p>

Integrability and chaotic dynamics in nonlinear dispersive wave model

期刊论文
发表日期：

2026年4月5日

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Ifrah Iqbal,
Hamood Ur Rehman

摘要

In this paper, we study a detailed analysis of the modified Benjamin-Bona-Mahony (BBM) equation, a nonlinear partial differential equation of significant interest in the investigation of nonlinear wave propagation and optical illusion modeling, wherein wave interference and chaotic interactions create intricate intensity distributions that are similar to optical patterns found in nonlinear optical systems. This equation, which generalizes traditional wave equations to include higher-order nonlinear and dispersive effects, is an effective tool for modeling intricate wave phenomena in a variety of physical systems. The collective impact of dispersive and nonlinear effects–where nonlinearity is amplitude-dependent self-interaction and phase modulation, and dispersion is for wave spreading and velocity dispersion. We start by attempting to test the integrability of the perturbed BBM equation through the Painlevé test. This technique, famous for proving the existence of movable singularities in solutions, validates the integrability characteristics of the equation and offers interesting information regarding the structure and nature of its solutions. Successful application of the Painlevé test proves that the equation has analytic, stable solutions under particular conditions. Next, we utilize the generalized exponential rational function approach to obtain a rich variety of exact wave solutions. This method, an extension of conventional solution methods, allows for the retrieval of multiple functional forms, such as solitary wave solutions, periodic wave solutions, and rational function solutions. Every solution uncovers unique physical properties of the inherent wave dynamics, leading to an enhanced understanding of the nonlinear phenomena described by the equation. To visualize and describe these solutions, we apply a variety of graphical methods. The solutions are displayed using 2D plots to show waveform profiles, 3D surface plots to expose amplitude change in space and time, revolution plots to show symmetry and geometrical structure, and density plots to record localized intensity variation. These visual representations offer an intuitive and complete view of the dynamic behavior produced by the solutions. Besides deriving and plotting solutions, we examine their stability in the context of a Hamiltonian system. Based on an analysis of the energy conservation and perturbation response of the system, we evaluate the resilience and long-term behavior of the solutions under small perturbations, providing information regarding their physical feasibility and engineering implications. We also examine the intricate and chaotic dynamics embedded in the system. With careful numerical computations, we point out phenomena like multistability, bifurcations, and initial value sensitivity. The appearance of chaotic dynamics is illustrated with a set of diagnostic methods, such as 2D and 3D phase portraits, time-series graphs, and Poincaré maps. These diagnostics illustrate the complex pathways in phase space and the uncertain but deterministic behavior of the system under different parameters and initial conditions. Overall, this research not only improves the theoretical framework of the modified BBM equation but also gives a solid set of analytical and computational tools to investigate nonlinear wave dynamics in applied settings. The originality of the work stems from the initial implementation of the generalized exponential rational function method (GERFM) on the mBBM equation to generate a variety of analytical solutions with detailed stability and chaos analysis going beyond conventional methods. These results show dynamically stable (Lyapunov-stable) and physically realistic solutions, obtained by choosing parameter ranges that guarantee bounded, nonsingular wave behavior and rule out complex or imaginary solutions.