Bifurcations, Chaos, Sensitivity Analysis, and Different Families of Soliton Solutions for a (2+1)-Dimensional Nonlinear Cubic Klein-Gordon Equation
-
Setu Rani [2] ; Sachin Kumar [1]
-
[1] University of Delhi
University of Delhi
India
-
[2] Lady Shri Ram College for Women
-
- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 2, 2026
- Idioma: inglés
- Enlaces
-
Resumen
This paper addresses the (2+1)-dimensional cubic Klein–Gordon (cKG) equation, which illustrates the dislocation propagation in crystals as well as the quantum behavior of elementary particles and appears in applications of mathematical physics, such as plasma physics, quantum field theory, and condensed matter physics. To explore analytical structures whose diversity is limited in existing studies, the new modified Sardar sub-equation method, Unified method and newly introduced generalised exponential rational function method are employed to obtain novel solutions in terms of hyperbolic, trigonometric, exponential, elliptic, and rational functions, as well as solutions involving arbitrary functions. The obtained solutions are visualized using 3D, 2D, and contour diagrams for relevant parameter values, effectively illustrating their physical significance and geometrical interpretation. These visualizations present a variety of wave forms, including periodic lump waves, multi-soliton, single-soliton, breather, rough wave, and peakons, along with their interactions. The main contribution of this paper lies in analyzing the bifurcation patterns, chaotic dynamics, and sensitivity characteristics of the governing equation. By reformulating the equation as a two-dimensional planar dynamical system, its behavior is extensively examined through qualitative analysis using phase portraits, and the response of solutions under various disturbances is discussed. Also, these plots are used to construct some new traveling wave solutions. Qualitative analysis of the system is demonstrated through sensitivity analysis, and a comparison with existing literature emphasizes the effectiveness of our work. Moreover, the modulation instability (MI) of the cKG equation is investigated, as it plays a significant role in characterizing the behavior of nonlinear wave propagation. The importance of the results lies in their ability to represent a broad range of complex and diverse phenomena encountered in both mathematical and physical systems.
-
Referencias bibliográficas
- 1. Alam, M.N.: Investigation of new solitary stochastic structures to the Heisenberg ferromagnetic spin chain model via a Stratonovich sense....
- 2. Hamid, I.,Kumar, S.: Newly formed solitarywave solutions and other solitons to the (3+ 1)-dimensional mKdV-ZK equation utilizing a...
- 3. Ghanbari, B., Inc, M.: A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear...
- 4. Kumar, S., Rani, S., Mann, N.: Diverse analytical wave solutions and dynamical behaviors of the new (2 + 1)-dimensional Sakovich equation...
- 5. Kumar, S., Rani, S.: Symmetries of optimal system, various closed-form solutions, and propagation of different wave profiles for the Boussinesq-Burgers...
- 6. Rafiq, M.H., Raza, N., Jhangeer, A.: Nonlinear dynamics of the generalized unstable nonlinear Schrödinger equation: a graphical perspective....
- 7. Zhang, R.F., Bilige, S.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations...
- 8. Osman, M.S., Machado, J.A.T., Baleanu, D.: On nonautonomous complex wave solutions described by the coupled Schrödinger-Boussinesq equation...
- 9. Shagolshem, S., Bira,B., Nagaraja, K.V.:Analysis of solitonwave structure for coupled Higgs equation via Lie symmetry, Paul Painlevé approach...
- 10. Kumar, S., Dhiman, S.K.: Exploring cone-shaped solitons, breather, and lump-forms solutions using the lie symmetry method and unified...
- 11. Refaie Ali, A., Alam,M., N., Fayz-Al-Asad,M. et al.: Precise soliton solutions of theLandau-Ginzburg- Higgs equation via the Sardar sub-equation...
- 12. Raza, N., Arshed, S.: Chiral bright and dark soliton solutions of Schrödinger’s equation in (1+2)- dimensions. Ain Shams Engineering...
- 13. Arshed, S., Raza, N., Alansari,M.: Soliton solutions of the generalized Davey-Stewartson equationwith full nonlinearities via three integrating...
- 14. Zubair, A., Raza, N., Mirzazadeh, M., Liu, W., Zhou, Q.: Analytic study on optical solitons in paritytime- symmetric mixed linear and...
- 15. Raza, N., Jhangeer, A., Ur Rahman, R., Butt, A.R., Chu, Y.-M.: Sensitive visualization of the fractionalWazwaz– Benjamin–Bona–Mahony equationwith...
- 16. Ashraf, M., Muhammad, N., Masud, M.A.B., Alam, M.N., Hossen, M.J.: Modeling and Analysis of Dynamic Waveforms in Nonlinear Fractional...
- 17. Ismael, H.F., Seadawy, A., Bulut, H.: Multiple soliton, fusion, breather, lump, mixed kink-lump and periodic solutions to the extended...
- 18. Ghanbari, B., Inc, M.: A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear...
- 19. Wang, M., Li, X., Zhang, J.: The G/G -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical...
- 20. Wu, F., Alam, M. N., Baskonus, H. M., Rezazadeh, H.: Closed-form wave solutions to the van der Waals model arising in nature and the longitudinal...
- 21. Hietarinta, J.: Introduction to the Hirota bilinear method. In:Kosmann-Schwarzbach,Y., Grammaticos, B., Tamizhmani, K.M. (eds) Integrability...
- 22. Kumar, S., Rani, S., Mann, N.: Diverse analytical wave solutions and dynamical behaviors of the new (2+ 1)-dimensional Sakovich equation...
- 23. Raza, N., Javid, A., Bulut, H.: Dynamics of optical solitons with the Radhakrishnan–Kundu– Lakshmanan model via two reliable integration...
- 24. Raza, N., Arshed, S., Javid, A.: Optical solitons and stability analysis for the generalized second-order nonlinear Schrödinger equation...
- 25. Ismael, H.F., Atas, S.S., Bulut, H., Osman, M.S.: Analytical solutions to the M-derivative resonant Davey-Stewartson equations. Mod. Phys....
- 26. Ismael, H.F., Ma,W.-X., Bulut, H.: Dynamics of soliton and mixed lump-solitonwaves to a generalized Bogoyavlensky-Konopelchenko equation....
- 27. Raza, N., Murtaza, I.G., Sial, S., Osman, M.S.: On solitons: the biomolecular nonlinear transmission line models with constant and time...
- 28. Coleman, S.: Quantum sine-Gordon equation as the massive Thirring model. Physical Review D 11(8), 2088 (1975)
- 29. Khan, K., Akbar, M.A.: Exact solutions of the (2+ 1)-dimensional cKG equation and the (3+ 1)- dimensional Zakharov-Kuznetsov equation...
- 30. Wu, G., Guo, Y., Yu, Y.: Nonlinear Complex Wave Excitations in (2+ 1)-Dimensional Klein-Gordon Equation Investigated by New Wave Transformation....
- 31. Wang, Z., Zhang, H.Q.: Many new kinds exact solutions to (2+ 1)-dimensional Burgers equation and Klein-Gordon equation used a new...
- 32. Seadawy, A.R., Lu, D.C., Arshad,M.: Stability analysis of solitary wave solutions for coupled and (2+ 1)-dimensional cubic Klein-Gordon...
- 33. Raza, N., Butt, A.R., Javid, A.: Approximate Solution of Nonlinear Klein-Gordon Equation Using Sobolev Gradients. Journal of Function...
- 34. Islam, M.S., Akbar, M.A., Khan, K.: Analytical solutions of nonlinear Klein-Gordon equation using the improved F-expansion method. Opt....
- 35. Alsisi, A.: Analytical and numerical solutions to the Klein-Gordon model with cubic nonlinearity. Alex. Eng. J. 99, 31–37 (2024)
- 36. Perko, L.: Differential equations and dynamical systems, third edition, Texts in Applied Mathematics, 7. Springer-Verlag, New York (2001)
- 37. Abdulkareem, H.H., Ismael, H.F.: Bifurcation and exact optical solutions in weakly nonlocal media with cubic-quintic nonlinearity. Nonlinear...
- 38. Raza, N., Jhangeer, A., Arshed, S., Butt, A.R., Chahlaoui, Y.: Dynamical analysis and phase portraits of two-mode waves in different media....
- 39. Byrd, P. F., Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, second edition, Springer, Berlin, Heidelberg,...
