Jacobi Elliptic Function Method for (2+1)-Dimensional Maccari System with Multiplicative Brownian Noise: Solitary Wave Solutions, Nonlinear Dynamics and Numerical Verification

• Peng Guo ^[1] ; Liran Wang ^[1] ; Jianming Qi ^[1]
  1. [1] Shanghai Dianji University

     Shanghai Dianji University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Introduction This paper focuses on the (2+1)-dimensional Maccari systemwith multiplicative Brownian noise, and conducts systematic research on the theoretical method, solution construction, nonlinear dynamics and numerical verification.

    Objective To address the lack of diversity in solitarywave solutions, nonlinear dynamical analysis and high-precision numerical verification of the stochastic system, and to establish a stochastic wave model consistent with the actual physical environments.

    Methodology The Jacobi Elliptic Functions (JEFs) method is adopted to solve the stochastic nonlinear partial differential system rather than the traditional extended tanh method, and the partial differential system is simplified to a system of ordinary differential equations through a traveling wave transformation. Nonlinear dynamical analysis is carried out by combining phase space trajectories, power spectral density and Lyapunov exponents. The adaptive Runge–Kutta method is used to numerically verify the Hamiltonian system.

    Main Findings The JEFsmethod can generate a large number of double-periodic elliptic function wave solutions, and construct various solitary wave solutions including parabolic, needle-shaped, periodic, bright and single breather waves. The noise intensity, wave velocity, and medium characteristic parameters can effectively regulate the evolution of waveform, which is consistent with real physical multi-factor coupling scenarios. Dynamical analysis reveals the topological structure and stability of equilibrium points, and the system shows no chaotic behavior when not subjected to external excitation, whereas chaotic behavior emerges when the system is under external forcing within the investigated parameter range. The numerical solution achieves high accuracy and strictly satisfies the energy conservation property of the Hamiltonian system.

    Significance This work establishes a complete research framework covering methodological innovation, solution construction and numerical verification, which breaks through the limitation of finite solution types in traditional methods for stochastic nonlinearwave equations. This study provides theoretical reference and numerical support for the analysis and application of stochastic nonlinear waves in plasma physics, nonlinear optics and fluid dynamics.

• Referencias bibliográficas
  • 1. Minati, L.: Across neurons and silicon: some experiments regarding the pervasiveness of nonlinear phenomena. Acta Phys. Pol. B 49, 2029–2093...
  • 2. Iqbal, I., Rehman, U.H., Alrashdi, T.: Fractional memory effects in nonlinear waves: dynamical study of solitons and chaotic attractors....
  • 3. Rehman, U.H., Jafar, N., Alrashedi, Y.: Fractional wave dynamics and solition solutions in Magneto- Electro-Elastic circular rods. Fractals....
  • 4. Iqbal, I., Rehman, U.H.: Fractional solitons and climate’s secret currents: Insights from the generalized fractional Kundu-Mukherjee-Naskar...
  • 5. Rahaman, M.S., Ullah, M.S., Islam, M.N.: Novel dynamics of the nonlinear fractional soliton neuron model with sensitivity analysis. AIP...
  • 6. Rahaman, S.M., Islam, N.M., Ullah, S.M.: Bifurcation, chaos, modulation instability, and soliton analysis of the schrödinger equation with...
  • 7. Murad, S.A.M., Hamasalh, F.K., Arnous, A.H., et al.: Optical solitons with conformable fractional evolution for the (3+1)-dimensional...
  • 8. Murad, S.A.M., Faridi, A.W., Jhangeer, A., et al.: Optical solutions with kudryashov’s arbitrary type of generalized non-local nonlinearity...
  • 9. Murad, S.A.M., Faridi, A.W., Jhangeer, A., et al.: The fractional soliton solutions and dynamical investigation for planer Hamiltonian...
  • 10. Butt, M.R.,Raza, N., Inc, M., et al.:Complexitons,Bilinear forms and Bilinear Backlund transformation of a (2+1)-dimensional Boiti-Leon-Manna-Pempinelli...
  • 11. Hosseini, K., Mirzazadeh, M., Baleanu, D., et al.: The generalized complex Ginzburg-Landau model and its dark and bright soliton solutions....
  • 12. Raza, N., Ullah, M.A.: A comparative study of heat transfer analysis of fractional Maxwell fluid by using Caputo and Caputo-Fabrizio derivatives....
  • 13. Chou, D., Amer, A., Rehman, H., et al.: Unravelling quiescent optical solitons: An exploration of the complex Ginzburg-Landau equation...
  • 14. Chou, D., Salah, M.B., Hamood, U.R., et al.: Additional investigation of the Biswas-Arshed equation to reveal optical soliton dynamics...
  • 15. Maged, F.A., Nauman, R., Muhammad, H.R., et al.: New solitary waves, bifurcation and chaotic patterns of Fokas system arising in monomode...
  • 16. Raza, N., Kaplan,M., Javid, A., et al.: Complexiton and resonant multi-solitons of a (4+1)-dimensional Boiti-Leon-Manna-Pempinelli...
  • 17. Chou, D., Iqbal, I., Alrashedi, Y., et al.: Bifurcation, chaotic behaviour, multistability and sensitivity analysis: Exact and numerical...
  • 18. Chou, D., Iqbal, I., Aithobaiti, T., et al.: Machine learning-assisted modeling of nonlinear dispersive waves in ocean engineering. Ocean...
  • 19. Nasreen, N., Seadawy, R.A., Lu, D.: Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using...
  • 20. Nasreen, N., Younas, U., Sulaiman, T.A., et al.: A variety of M-truncated optical solitons to a nonlinear extended classical dynamical...
  • 21. Nasreen, N., Seadawy, R.A., Lu, D., et al.: Optical fibers to model pulses of ultrashort via generalized third-order nonlinear Schrödinger...
  • 22. Wang, J.L., He, J.S.: The distortion study of rogue waves of the generalized nonlinear Schrödinger equation under the third-order dispersion...
  • 23. Wang, R., Han, L., Zhang, M.N., et al.: Nonlinear distortion of nonreciprocal transmission in paritytime- symmetric silicon micromechanical...
  • 24. Davies, O.A., Masood, C.K.: Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional...
  • 25. Akram, S., Rahman, U.M., Essa, A.A.L.: Comprehensive analysis of a magneto-electro-elastic longitudinal wave model: Symmetries, chaos,...
  • 26. Bing, L.X., Duan, S.W., Li, Z.Z.: Emergence of the dark envelope nonlinear surface waves in shallow water. Eur. Phys. J. Spec. Top. 234,...
  • 27. Magoulès, F., Nguyen, T.P.K., Omnes, P., et al.: Optimal absorption of acoustic waves by a boundary. SIAM J. Control. Optim. 59, 561–583...
  • 28. Rafiq, N.M., Rafiq, H.M., Alsaud, H.: New insights into the diversity of stochastic solutions and dynamical analysis for the complex cubic...
  • 29. Bo, Y., Tian, D., Liu, X., et al.: Discrete maximum principle and energy stability of the compact difference scheme for two-dimensional...
  • 30. Xu, J., Jiang, Y., Wang, N., et al.: Progress in local-variable-based transition-turbulence models for subsonic and transonic boundary...
  • 31. Zhou, Y.Q., Li, J.B., Zhang, Q.Y.: Bifurcations and exact solutions for the nonlinear schrödinger equation in a nonlinear saturable medium....
  • 32. Tuan, N., Thach, T.: Rayleigh-Stokes equations with space-time white noise: Existence, Hölder regularity, and parameter-continuity. Potential...
  • 33. Arif, K., Xi, G., Wang, H., et al.: Deep learning for high-dimensional PDEs with fat-tailed Lévy measure. J. Comput. Phys. 541, 114327...
  • 34. Guo, J., Song, Y., Wang, Z., et al.: A hybrid data-driven prognostic scheme based on unsupervised health indicator construction and random-effects...
  • 35. Ma, X., Xiao, C., Meng, Q.: Open-loop and closed-loop saddle points of infinite dimensional linearquadratic stochastic differential games...
  • 36. Migliorini, G., Fanelli, D.: Multiplicative noise induced bistability and stochastic resonance. J. Phys. Complex. 5, 045001 (2024)
  • 37. Ghasemi, S.E.: Numerical study of non-newtonian nanofluid flow influenced by thermophoresis and Brownian motion. Res. in Eng. 28, 107701...
  • 38. Wang, Q., Zu, L., Jiang, D.: Rich dynamics and density function analysis of a hybrid stochastic predator-prey model with logistic growth...
  • 39. Kalpana, G., Madhura, K.R., Kudenatti, R.B.: Numerical study on the combined effects of Brownian motion and thermophoresis on an unsteady...
  • 40. Schilling, L.R.: Brownian Motion:AGuide to Random Processes and Stochastic Calculus. De Gruyter, Berlin (2021)
  • 41. Zhong, M.,Weng,W.,Guo, B., et al.:The fractional nonlinear Schrödinger equation: Soliton turbulence, modulation instability, and extreme...
  • 42. Alharbi, Y.F., Sohaly, M.A., Abdelrahman, M.A.E.: New stochastic solutions for a new extension of nonlinear Schrödinger equation. Pramana-J....
  • 43. Maccari, A.: The Kadomtsev-Petviashvili equation as a source of integrable model equations. J.Math. Phys. 37, 6207–6212 (1996)
  • 44. Wang, G.H., Wang, L.H., Rao, J.G., et al.: New patterns of the two-dimensional rogue waves: (2+1)- dimensional Maccari system. Commun....
  • 45. Xu, T., Chen, Y., Qiao, Z.: Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system. Mod. Phys. Lett....
  • 46. Alomair, R.A., Hassan, S.Z., Abdelrahman, M.A.E.: A new structure of solutions to the coupled nonlinear Maccari system in plasma physics....
  • 47. Yousaf, M.Z., Abbas, M., Abdullah, F.A., et al.: Construction of travelling wave solutions of coupled Higgs equation and the Maccari system...
  • 48. Qing, B.L., Majid, A.W., Lan, Y.M.: Soliton resonances, soliton molecules to breathers, semi-elastic collisions and soliton bifurcation...
  • 49. Zhao, Y., Tian, B., Hu, C.C., et al.: Interactions among the rogue waves and solitons for a (2+1)- dimensionalMaccari system in fluid...
  • 50. Schilling, R. L.: Stochastic differential equations: an introduction with applications (6th edn), by Bernt Oksendal. Math. Gaz. 89, 178-179...
  • 51. Kloeden, P.E., Pearson, R.A.: Numerical solution of stochastic differential equations. J. Aust. Math. Soc. 20, 8 (1992)
  • 52. Zhao, W., Liu, Y.J., Liu, L.: Observer-Based adaptive Fuzzy tracking control using integral barrier Lyapunov functionals for a nonlinear...
  • 53. Giuseppe, P., Austin, M., Scott, H., et al.: Stratonovich-to-Itô transition in noisy systems with multiplicative feedback. Nat. Commun....
  • 54. Faiqa, A., Adil, J., Muhammad, M., et al.: Solitonic, quasi-periodic, super nonlinear and chaotic behaviors of a dispersive extended nonlinear...
  • 55. Muhammad, H.R., Adil, J.,Nauman, R.: Symmetry and complexity:ALie symmetry approach to bifurcation, chaos, stability and travelling wave...
  • 56. Beenish, Harun, K., Muhammad, B.R., et al.: Lie analysis and nonlinear propagating waves of the (3+1)-dimensional generalized Boiti-Leon-Manna-Pempinelli...
  • 57. Zakarya, M., Shehri, A.Z.N., Ali, M.H., et al.: Analytical solutions for generalized stochastic HSCKdV equations with variable coefficients...
  • 58. Murad, S.A.M.,Younas, U., Emadifar, H., et al.:Diverse soliton solutions to the conformable Ivancevic option pricing model via the modified...
  • 59. Hussain, A., Kara, A.H., Zaman, F.D.: Symmetries, associated first integrals and successive reduction of Schrödinger type and other second...
  • 60. Alharthi, M.: The modulations of collapsing stochastic (2+1)-dimensional Maccari system structures. Alex. Eng. J. 129, 738–743 (2025)
  • 61. Tuffour, F., Barnes, B., Dontwi, K.I., et al.: The exact solution of the fractional Burger’s equation using the modified homogeneous balance...
  • 62. Silambarasan, R., Kilicman, A., Jyotheeswari, P.: The Jacobi elliptic function and incomplete elliptic integral of second kind solutions...
  • 63. Kumar, P.D.: Superposed and superposed-type double periodic Jacobi elliptic function solutions of variable coefficients KdV equation....
  • 64. Ahmed, H.K., Ali, K.K.: Optical solutions to the stochastic Fokas-Lenells equation with multiplicative white noise in Itô sense using...
  • 65. Liu, S., Fu, Z., Liu, S., et al.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys....
  • 66. Wang, M.L., Li, X.Z.: Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos Solitons...
  • 67. Silambarasan, R., Nisar, S.K., Jyotheeswari, P.: Jacobi elliptic patterns and compound non-topological solitons of (2+1) shallow water...
  • 68. Wang, X., He, J.: Dark solitons on elliptic function background for the defocusing Hirota equation. Commun. Theor. Phys. 77, 35003 (2025)
  • 69. Rafiq, M.H., Raza, N., Jhangeer, A.: Nonlinear dynamics of the generalized unstable nonlinear Schrödinger equation: a graphical perspective....
  • 70. Hamza, M.R., Adil, J., Nauman, R.: The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns...
  • 71. Tabekoueng, Z.N., Balamurali, R., Karthikeyan, R., et al.: Extremely rich dynamics of coupled heterogeneous neurons through a Josephson...
  • 72. Hirsch,M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press (2004)
  • 73. Kumar, S., Rani, S., Mann, N.: Analytical soliton solutions to a (2+1)-dimensional variable coefficients graphene sheets equation...
  • 74. McDonald, E.J., Higham, D.J.: Error analysis of QR algorithms for computing Lyapunov exponents. Electron. T. Numer. Ana. 12, 234–251 (2001)
  • 75. Hairer, E., Nørsett, S.P.,Wanner, G.: SolvingOrdinaryDifferential Equations I. Springer, Berlin (2006)
  • 76. Cheemaa, N.,Younis, M.:Newandmore exact travelingwave solutions to integrable (2+1)-dimensional Maccari system. Nonlinear Dynam. 83,...
  • 77. Sachin, K., Setu, R.: Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+1)-dimensional...