The traveling wave equation of the generalized nonlinear dispersive Boussinesq equation is a singular system with two possible singularities, which is transformed into a regular system after making a scale transformation. The bifurcations and phase portraits of the regular system are obtained by using bifurcation theory of dynamical systems. The singular traveling wave theory is used to discuss the impact of singularity on the smoothness of solutions and analyze the reason for the appearance of non-smooth solutions such as soliton cusp waves and periodic cusp waves. Under different parameter conditions, the sufficient conditions for the existence of smooth and non-smooth traveling wave solutions of the singular system are given. For certain special cases, the parametric representations of 25 explicit and exact traveling wave solutions are presented such as smooth peak-shaped, valley-shaped, and kink-shaped soliton waves along with non-smooth soliton cusp waves. Furthermore, 2D wave plots of smooth soliton waves and non-smooth soliton cusp wave solutions are drawn to visualize the dynamics of the equation. The results of this study contribute to the better understanding of the generalized nonlinear dispersive Boussinesq equation.

generalized nonlinear dispersive Boussinesq equation, bifurcation, phase portrait, soliton cusp wave, periodic cusp wave.

Received: April 10, 2024; Accepted: June 1, 2024; Published: July 4, 2024

How to cite this article: Dahe Feng, Airen Zhou and Jianjun Jiao, Bifurcations, smooth and non-smooth traveling wave solutions for generalized nonlinear dispersive Boussinesq equation, Far East Journal of Dynamical Systems 37(2) (2024), 137-177. https://doi.org/10.17654/0972111824007

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] Y. Li, R. Yao and S. Lou, An extended Hirota bilinear method and new wave structures of -dimensional Sawada-Kotera equation, Appl. Math. Lett. 145 (2023), 108760.
[2] L. Li, C. Duan and F. Yu, An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation, Phys. Lett. A 383(14) (2019), 1578-1582.
[3] Y. Yang, T. Suzuki and J. Wang, Bäcklund transformation and localized nonlinear wave solutions of the nonlocal defocusing coupled non-linear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 95 (2021), 105626.
[4] Y. Yang, T. Xia and T. Liu, Darboux transformation and exact solution to the nonlocal Kundu-Eckhaus equation, Appl. Math. Lett. 141 (2023), 108602.
[5] H. Chen and S. Zheng, Darboux transformation for nonlinear Schrödinger type hierarchies, Phys. D 454 (2023), 133863.
[6] G. Iskenderoglu and D. Kaya, Chirped self-similar pulses and envelope solutions for a nonlinear Schrödinger’s in optical fibers using Lie group method, Chaos Sotiton. Fract. 162 (2022), 112453.
[7] M. Moustafa, A. M. Amin and G. Laouini, New exact solutions for the nonlinear Schrödinger’s equation with anti-cubic nonlinearity term via Lie group method, Optik 248 (2021), 168205.
[8] S. Rekha, R. R. Rani, L. Rajendran and M. E. G. Lyons, A new method to study the nonlinear reaction-diffusion process in the electroactive polymer film using hyperbolic function method, Int. J. Electrochem. Sci. 17(12) (2022), 221261.
[9] H. U. Rehman, A. U. Awan, E. M. Tag-ElDin, S. E. Alhazmi, M. F. Yassen and R. Haider, Extended hyperbolic function method for the -dimensional nonlinear soliton equation, Results Phys. 40 (2022), 105802.
[10] S. Mostafa, R. El-Barkouky, H. M. Ahmed and I. Samir, Investigation of chirped optical solitons perturbation of higher order NLSE via improved modified extended tanh function approach, Results Phy. 52 (2023), 106760.
[11] E. M. E. Zayed, M. E. M. Alngar and M. El-Horbaty, Optical solitons in fiber Bragg gratings having Kerr law of refractive index with extended Kudryashov’s method and new extended auxiliary equation approach, Chinese J. Phys. 66 (2020), 187-205.
[12] R. Silambarasan and K. S. Nisar, Doubly periodic solutions and non-topological solitons of -dimension Wazwaz Kaur Boussinesq equation employing Jacobi elliptic function method, Chaos Sotiton. Fract. 175(1) (2023), 113997.
[13] A. Hussain, Y. Chahlaoui, F. D. Zaman, T. Parveen and A. M. Hassan, The Jacobi elliptic function method and its application for the stochastic NNV system, Alex. Eng. J. 81 (2023), 347-359.
[14] F. D. Emmanuel, W. K. T. Gildas, I. D. Zacharie, K. J. Aurélien, P. N. Jean and N. Laurent, Exotical solitons for an intrinsic fractional circuit using the sine-cosine method, Chaos Sotiton. Fract. 160 (2022), 12253.
[15] S. W. Yao, S. Behera, M. Inc, H. Rezazadeh, J. P. S. Virdi, W. Mahmoud, O. A. B. Arqub and M. S. Osman, Analytical solutions of conformable Drinfel’d-Sokolov-Wilson and Boiti Leon Pempinelli equations via sine-cosine method, Results Phy. 42 (2022), 105990.
[16] A. K. M. K. S. Hossain and M. A. Akbar, Traveling wave solutions of Benny Luke equation via the enhanced -expansion method, Ain Shams Eng. J. 12(4) (2021), 4181-4187.
[17] L. Zhang, L. Chen and X. Huo, New exact compaction, peakon and solitary solutions of the generalized Boussinesq-like equations with nonlinear dispersion, Nonlinear Anal.-Theor. 67(12) (2007), 3276-3282.
[18] W. Sun and Y. Sun, The degenerate breather solutions for the Boussinesq equation, Appl. Math. Lett. 128 (2022), 107884.
[19] Z. Yan, Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation. II: Solitary pattern solutions, Chaos Soliton Fract. 18 (2003), 869-880.
[20] A. M. Wazwaz, Solutions of compact and noncompact structures for nonlinear Klein-Gordon-type equation, Appl. Math. Comput. 134(2-3) (2003), 487-500.
[21] D. Feng, J. Li and J. Jiao, Dynamical behavior of singular traveling waves of -dimensional nonlinear Klein-Gordon equation, Qual. Theor. Dyn. Syst. 18 (2019), 265-287.
[22] S. Chow and J. K. Hale, Method of Bifurcation Theory, Springer-Verlag, New York, 1981.
[23] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
[24] J. Li and Z. Liu, Traveling wave solutions for a class of nonlinear dispersive equations, Chinese Ann. Math. B 23(3) (2003), 397-418.