The Burgers equation, KdV equation and Burgers-KdV equation are real physical modes concerning many branches in physics. In this paper, rich types of explicit and exact travelling wave solutions for these three equations, including the solitary wave solutions, the singular travelling wave solutions, the triangle function periodic wave solutions, etc., are presented by a direct trial function approach. Among them, some are new travelling wave solutions.

nonlinear wave equation, direct trial function approach, solitary wave solution, singular travelling wave solution, triangle function solution

Received: February 10, 2025; Accepted: March 10, 2025; Published: March 20, 2025

How to cite this article: Yuan-Xi Xie, Solutions to a class of nonlinear wave equations, Far East Journal of Applied Mathematics 118(1) (2025), 19-38. http://dx.doi.org/10.17654/0972096025003

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

References:
[1] P. K. Shukla, N. N. Rao and M. Y. Yu, Relativistic nonlinear effects in plasmas, Phys. Rep. 138 (1986), 1-149.
[2] M. J. Alowitz and H. Segur, Solitons and the Inverse Scattering Transforms, Philadelphia, SIAM, 1981.
[3] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys. 60 (1992), 650-654.
[4] E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000), 212-218.
[5] S. K. Liu, Z. T. Fu and S. D. Liu, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289 (2001), 69-74.
[6] E. J. Parkes, B. R. Duffy and P. C. Abbott, The Jacobi elliptic function method for finding periodic-wave solutions to nonlinear evolution equations, Phys. Lett. A 295 (2002), 280-286.
[7] Y. X. Xie and J. S. Tang, A unified trial function method in finding the explicit and exact solutions to three NPDEs, Phys. Scr. 74 (2006), 197-200.
[8] Y. X. Xie, Explicit and exact solutions to the KdV-Burgers equation, Nuovo Cimento B 121 (2006), 689-697.
[9] Y. X. Xie, Explicit and exact solutions to the mKdV-sine-Gordon equation, Mod. Phys. Lett. B 22 (2008), 1471-1485.
[10] Y. X. Xie, A combination method and its applications to nonlinear evolution equations, Int. J. Mod. Phys. B 26 (2012), 1250110-1-1250110-10.
[11] Y. X. Xie, Seeking the explicit exact solutions of sinh-Poisson type equations, Far East J. Appl. Math. 87 (2014), 159-180.
[12] C. P. Liu, The relation between the kink-type solution and the kink-bell-type solution of nonlinear evolution equations, Phys. Lett. A 312 (2003), 41-48.
[13] Z. T. Fu, S. D. Liu and S. K. Liu, Notes on solutions to Burgers-type equations, Commun. Theor. Phys. 41 (2004), 527-530.
[14] S. F. Shen and Z. L. Pan, A note on the Jacobi elliptic function expansion method, Phys. Lett. A 308 (2003), 143-148.