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

• Ayesha Ejaz ^[1] ; Nauman Raza ^[4] ; Younes Chahlaoui ^[2] ; Mustafa Bayram ^[3] ; Ahmet Bekir ^[5]
  1. [1] University of the Punjab

     University of the Punjab

     Pakistán

     
  2. [2] King Khalid University

     King Khalid University

     Arabia Saudí

     
  3. [3] Biruni University

     Biruni University

     Turquía

     
  4. [4] University of the Punjab, Khazar University
  5. [5] Imarli Street

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • TheBenjamin-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 theGrassberger–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.

• Referencias bibliográficas
  • 1. Zaman, U.H., Arefin, M.A., Akbar, M.A., Uddin, M.H.: Study of the soliton propagation of the fractional nonlinear type evolution equation...
  • 2. Guo, Y., Cao, X., Peng, K.: Solving nonlinear soliton equations using improved physics-informed neural networks with adaptive mechanisms....
  • 3. Raza, N., Rafiq,M.H., Bekir, A., Rezazadeh, H.: Optical solitons related to (2+1)-dimensional Kundu– Mukherjee–Naskar model using an...
  • 4. Wang, X., Ansar, R., Abbas, M., Abdullah, F.A., Abualnaja, K.M.: The investigation of dynamical behavior of Benjamin–Bona–Mahony–Burgers...
  • 5. Kumari, A.,Kukreja,V.K.: Septic Hermite collocationmethod for the numerical solution of Benjamin– Bona–Mahony–Burgers equation. J. Differ....
  • 6. Akcagil, S.,Aydemir, T.,Gozukizil, O.F.: Exact travellingwave solutions of nonlinear pseudoparabolic equations by using the (G’/G)- expansion...
  • 7. Gözükızıl ÖF, Akçail ¸ S: The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions. Advances in Difference...
  • 8. Kumar, S., Niwas, M.,Wazwaz, A.M.: Lie symmetry analysis, exact analytical solutions and dynamics of solitons for (2+1)-dimensional...
  • 9. Hussain, A., Usman, M., Al-Sinan, B.R., Osman,W.M., Ibrahim, T.F.: Symmetry analysis and closedform invariant solutions of the nonlinear...
  • 10. Zhang, J.L., Wang, M.L., Wang, Y.M., Fang, Z.D.: The improved F-expansion method and its applications. Phys. Lett. A 350(1–2), 103–109...
  • 11. Zhang, J.L., Wang, M.L., Wang, Y.M., Fang, Z.D.: The improved F-expansion method and its applications. Phys. Lett. A 350(1–2), 103–109...
  • 12. Yin, Y.H., Lü, X., Ma, W.X.: Bäcklund transformation, exact solutions and diverse interaction phenomena to a (3+1)-dimensional nonlinear...
  • 13. Cherniha, R.: Comments on the paper Exact solutions of nonlinear diffusion-convection-reaction equation: a Lie symmetry approach. Commun....
  • 14. Ibrahim, S., Ashir, A.M., Sabawi, Y.A., Baleanu, D.: Realization of optical solitons from nonlinear Schrödinger equation using modified...
  • 15. Kamel, N.M., Ahmed, H.M., Rabie, W.B.: Retrieval of soliton solutions for 4th-order (2+1)- dimensional Schrödinger equation with higher-order...
  • 16. Ahmad, J., Hameed, M., Mustafa, Z., Rehman, S.U.: Soliton patterns in the truncated M-fractional resonant nonlinear Schrödinger equation...
  • 17. Murad, M.A., Faridi, W.A., Jhangeer, A., Iqbal, M., Arnous, A.H., Tchier, F.: The fractional soliton solutions and dynamical investigation...
  • 18. Rani, S., Kumar, S.: Dynamics of soliton solutions and various evolving formations of the Jaulent- Miodek and Zakharov-Kuzetsov equations...
  • 19. Matouk, A.E., Lahcene, B.: Chaotic dynamics in some fractional predator–prey models via a new Caputo operator based on the generalised...
  • 20. Baker, G.L., Gollub, J.P.: Chaotic dynamics: An introduction, Cambridge University Press (1996)
  • 21. Farooq, F.B., Raza, N., Ijaz, A., Bayram, M., Vivas-Cortez, M.: A comprehensive analytical study of solitons and nonlinear dynamics in...
  • 22. Raza, N., Seadawy, A.R., Kaplan, M., Butt, A.R.: Symbolic computation and sensitivity analysis of nonlinear Kudryashov’s dynamical equation...
  • 23. Rafiq,M.H., Jhangeer, A., Raza, N.: Symmetry and complexity: A Lie symmetry approach to bifurcation, chaos, stability and travelling wave...
  • 24. Kumar, S., Rani, S., Mann, N.: Diverse analytical wave solutions and dynamical behaviors of the new (2+1)-dimensional Sakovich equation...
  • 25. Crawford, J.D.: Introduction to bifurcation theory. Rev. Mod. Phys. 63(4), 991 (1991)
  • 26. Jhangeer, A., Raza, N., Ejaz, A., Rafiq, M.H., Baleanu, D.: Qualitative behavior and variant soliton profiles of the generalized P-type...
  • 27. Strogatz, S.H.: Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering,...
  • 28. Riaz, M.B., Jhangeer, A., Martinovic, J., Kazmi, S.S.: Dynamics and soliton propagation in a modified Oskolkov equation: Phase plot insights....
  • 29. Benkouider, K., Bouden, T., Sambas, A., Lekouaghet, B., Mohamed, M.A., Ibrahim Mohammed, S., Mamat, M., Ibrahim,M.A., Ahmad, M.Z.: A new...
  • 30. Samuilik IN: Lyapunov exponents and Kaplan–Yorke dimension for five-dimensional system.WSEAS Transactions on Systems, 268-75 (2022)
  • 31. Bouke, M.A.: Fractal attractors in random nonlinear iterated function systems: Existence, stability, and dimensional properties, (2025)....
  • 32. Peitgen, H.O., Jürgens, H., Saupe, D., Feigenbaum, M.J.: Chaos and fractals: new frontiers of science 106, 560–604. Springer, New York...
  • 33. Grassberger, P., Procaccia, I.:Measuring the strangeness of strange attractors. Physica D 9(1–2), 189– 208 (1983)
  • 34. Tariq, K.U., Jhangeer, A., Ali, M.N., Ilyas, H., Tufail, R.N.: Lumps, solitons, modulation instability and stability analysis for the...
  • 35. Raza, N., Jhangeer, A., Amjad, Z., Rani, B., Baleanu, D.: Exploring soliton dynamics and wave interactions in an extended Kadomtsev-Petviashvili-Boussinesq...
  • 36. Jhangeer, A.,Talafha, A.M., Ansari, A.R.: Analytical solutions and dynamical behaviors of the extended Bogoyavlensky-Konopelchenko equation...
  • 37. Jhangeer,A.,Rˇ íha,L.: Symmetry analysis, dynamical behavior, and conservation laws of the dual-mode nonlinear fluid model. Ain Shams...
  • 38. Nadeem, M., Arqub, O.A.: Soliton solutions and stability analysis to the Hirota-bilinear-like model using the unified Riccati equation...
  • 39. Nadeem, M., Arqub, O.A., Alotaibi, F.M.: Optical soliton solutions and modulation instability for unstable conformable Schrödinger model....
  • 40. Nadeem, M., Jabeen, S., Arqub, O.A.: Investigating the invariant solutions of (1+1)-dimensional Sawada-Kotera model using Lie symmetries...
  • 41. Nadeem, M., Hayat, T.: Analyzing the bifurcation, chaos and soliton solutions to (3+1)-dimensional nonlinear hyperbolic Schrödinger...
  • 42. Wiggins, S.: Global bifurcations and chaos: Analytical methods. Springer-Verlag, Applied Mathematical Sciences 73, 495 (1988)
  • 43. Ullah,M.S., Ali, M.Z., Roshid, H.O.: Bifurcation, chaos, and stability analysis to the second fractional WBBM model. PLoS ONE 19(7), e0307565...
  • 44. Dhakshinamoorthy, V., Wu, G.C., Banerjee, S.: Chaotic dynamics of fractional discrete time systems, CRC Press (2024)
  • 45. Ejaz, A., Amjad, Z., Raza, N., Wong, P.J., Almalki, Y.: Dynamical observations and range of diverse soliton profiles for a nonlinear double-chain...
  • 46. Rani, S., Kumar, S., Mann, N.: On the dynamics of optical soliton solutions, modulation stability, and various wave structures of a (2+...
  • 47. Kumar, S., Rani, S.: Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+ 1)-dimensional...
  • 48. Rani, S., Kumar, S., Kumar, R.: Dynamical study of newly created analytical solutions, bifurcation analysis, and chaotic nature of the...
  • 49. Kumar, S., Rani, S., Ma, W.X.: Lie symmetries, modulation instability, conservation laws, and the dynamic waveform patterns of several...