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<p>The Benjamin-Bona-Mahony-Peregrine-Burgers equation characterizes nonlinear dispersive and dissipative processes that are applicable to traffic flow and shallow-water waves. In order to find symmetry generators and convert the PDE to ODE form, this approach starts with a Lie symmetry analysis. The modified Sardar sub-equation approach is then used to construct exact traveling wave solutions, which are then displayed using 2D, density, and 3D plots. The dynamics of the system are investigated in two phases. First, bifurcation analysis of the corresponding planar model demonstrates how parameter variation affects stability. Second, phase portraits, return maps studied by the Grassberger–Procaccia algorithm, and Poincar’e maps with fractal dimensions derived by the box-counting approach are used to examine chaotic behavior for a perturbed version of the system. The power spectrum, bifurcation diagrams, Lyapunov exponents, and the Kaplan-Yorke dimension provide more proof of chaos. The coexistence of different attractors for the same parameters is finally shown using a multistability analysis. Together, these findings offer a coherent picture of the exact solutions, stability patterns, and nonlinear dynamical characteristics of the BBMPB problem</p>

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Analytical Derivations of Traveling Waves, Parameter-Dependent Bifurcations, and Chaotic Dynamics in the BBMP–Burgers Equation

  • 期刊论文

  • 发表日期：

    2026年6月26日

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  • Ayesha Ejaz,
  • Nauman Raza,
  • Younes Chahlaoui,
  • Mustafa Bayram,
  • Ahmet Bekir

摘要

The Benjamin-Bona-Mahony-Peregrine-Burgers equation characterizes nonlinear dispersive and dissipative processes that are applicable to traffic flow and shallow-water waves. In order to find symmetry generators and convert the PDE to ODE form, this approach starts with a Lie symmetry analysis. The modified Sardar sub-equation approach is then used to construct exact traveling wave solutions, which are then displayed using 2D, density, and 3D plots. The dynamics of the system are investigated in two phases. First, bifurcation analysis of the corresponding planar model demonstrates how parameter variation affects stability. Second, phase portraits, return maps studied by the Grassberger–Procaccia algorithm, and Poincar’e maps with fractal dimensions derived by the box-counting approach are used to examine chaotic behavior for a perturbed version of the system. The power spectrum, bifurcation diagrams, Lyapunov exponents, and the Kaplan-Yorke dimension provide more proof of chaos. The coexistence of different attractors for the same parameters is finally shown using a multistability analysis. Together, these findings offer a coherent picture of the exact solutions, stability patterns, and nonlinear dynamical characteristics of the BBMPB problem