ANALYTICAL SOLUTIONS OF THE (2+1)-DIMENSIONAL GENERALIZED BENJAMIN-ONO EQUATION

The (2+1)-dimensional generalized Benjamin-Ono equation models the propagation of small-amplitude, long-wavelength waves on the surface of shallow water. Constructing explicit solutions of the (2+1)-dimensional generalized Benjamin-Ono equation not only provides theoretical support for experimental investigations but also offers a rigorous basis for addressing applied problems arising in nonlinear wave dynamics. In this work, we investigate wave propagation governed by the (2+1)-dimensional generalized Benjamin-Ono equation in nonlinear media, accounting for dispersive effects. To this end, the sine-cosine function method and the hyperbolic tangent method are employed as analytical tools for deriving explicit solutions. The methods prove effective for a broad class of nonlinear equations encountered in mathematical physics. Using these approaches, periodic-wave solutions and solitary wave solutions are obtained, and to illustrate the obtained results, we plot 3D and 2D plots by setting suitable values of the involved parameters.

1. Wazwaz, A.M., Partial Differential Equations and Solitary Waves Theory (Springer-Verlag, Berlin Heidelberg, 2009), 681 p.

2. Burdik, C., Shaikhova. G., Rakhimzhanov, B. Soliton solutions and travelling wave solutions for the two-dimensional generalized nonlinear Schrödinger equations, European Physical Journal Plus 136(10), 1095 (2021).

3. Manikandan, K., Serikbayev, N., Manigandan, M., Sabareeshwaran, M. Dynamical evolutions of optical smooth positons in variable coefficient nonlinear Schrödinger equation with external potentials. Optik 288, 171203 (2023).

4. Verhulst F. Nonlinear Differential Equations and Dynamical Systems (Springer, Berlin, 1996).

5. Shaikhova, G., Kutum, B., Myrzakulov, R.: Periodic traveling wave, bright and dark soliton solutions of the (2+1)-dimensional complex modified Korteweg-de Vries system of equations by using three different methods. AIMS Mathematics, 7(10), 18948–18970 (2022).

6. Chen, H., Zheng, Sh. Darboux transformation for nonlinear Schrödinger type hierarchies. Physica D: Nonlinear Phenomena, 454, 133863б (2023).

7. Liu, J.G., Zhu, W.H. Multiple rogue waves, breather wave and interaction solutions of a generalized (3+1)-dimensional variable-coefficient nonlinear wave equation. Nonlinear Dynamics, 103, 1841–1850 (2021).

8. Dehghan, M., Hamidi, A., Shakourifar, M. The solution of coupled Burgers, equations using AdomianPade technique. Applied Mathematics and Computation. 189(2), 1034–1047 (2007).

9. Cherniha, R., Muzyka, L. Lie symmetries of the Shigesada-Kawasaki-Teramoto system. Communication Nonlinear Science and Numerical Simulation, 45, 81–92 (2017).

10. Hosseini, K., Mayeli, P., Kumar ,D. New exact solutions of the coupled sine-Gordon equations in nonlinear optics using the modified Kudryashov method. Journal Modern Optics, 65(3), 361–364 (2018).

11. Yao Sh.-W, Behera. S., Inc, M., Rezazadeh, H., Virdi, J.P.S., Mahmoud, W., Arqu, O.A., Osman, M.S. Analytical solutions of conformable Drinfel’d-Sokolov-Wilson and Boiti Leon Pempinelli equations via sinecosine method. Results in Physics, 42, 105990 (2022).

12. Wazwaz, A.M., The sine-cosine method for obtaining solutions with compact and noncompact structures, Applied Mathematics and Computation, 159(2), 559–576 (2004).

13. Ala, V., Shaikhova, G. Analytical Solutions of Nonlinear Beta Fractional Schrödinger Equation Via Sine-Cosine Method, Lobachevskii Journal of Mathematics, 43(11), 3033–3038 (2022).

14. Malfliet, W. Solitary wave solutions of nonlinear wave equations, American Journal of Physics., 60 (7), 650–654 (1992).

15. Malfiiet, W. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta, 54, 563–568 (1996).

16. Wazwaz, A.M. The Tanh and the Sine-Cosine Methods for the Complex Modified KdV and the Generalized KdV Equations. Computers and Mathematics with Applications, 49, 1101–1112 (2005)

17. Hassaballa, A., Baber, M.Z., Butt, A., Yasin, M.W., Safdar, R., Bittaye, E. Dynamical description and analytical study of traveling wave solutions for generalized Benjamin-Ono equation. Scientific Reports, 15, 33923 (2025).

18. Ma, H., ·Yue, Sh., · Gao, Y., Deng, A. Lump solution, breather soliton and more soliton solutions for a (2+1)-dimensional generalized Benjamin-Ono equation. Qualitative Theory of Dynamical Systems, 22, 72 (2023).