Explicit and accurate solutions for the Benney equation

Yuan-Xi  Xie

doi:10.59400/jam2899

Explicit and accurate solutions for the Benney equation

Yuan-Xi Xie School of Physics and Electronic Science, Hunan Institute of Science and Technology, Yueyang 414000, China

Article ID: 2899

DOI: https://doi.org/10.59400/jam2899

Keywords: Benney equation; direct trial function approach; traveling wave solution; solitary wave solution

Abstract

The Benney equation arises from many different physical contexts as an appropriately real physical model equation involving a lot of effects of dispersion, dissipation, nonlinearity, and instability. As a result, it is a very important and challenging theme to search for the explicit and accurate traveling wave solutions of the Benney equation. In this paper, by introducing an ansatz solution with two E-exponential functions, we have made some improvements to the trial function approach for solving three NPDEs proposed by Xie and Tang. On this basis, we have put forward a direct trial function approach to search for the explicit and accurate traveling wave solutions of NEEs. We have demonstrated its effectiveness by applying it to the Benney equation. Therefore, a series of more general explicit and accurate traveling wave solutions to the Benney equation, comprising the solitary wave solutions and the singular traveling wave solutions, are successfully derived in a forthright and concise way. The obtained results are completely consistent with those given in the existing references. In addition, compared with the proposed approaches in the existing references, the technique described herein seems to be less calculative. Our approach may provide a novel way of thinking for solving NEEs. We firmly believe that the method used herein may also be applied to search for the explicit and accurate traveling wave solutions to other NEEs. We plan to extend this technique to search for the explicit and accurate traveling wave solutions of other NEEs.

References

[1]Malfliet W. Solitary wave solutions of nonlinear wave equations. American Journal of Physics. 1992; 60(7): 650-654. doi: 10.1119/1.17120

[2]Fan EG. Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A. 2000; 277: 212-218.

[3]Liu SK, Fu ZT, Liu SD. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A. 2001; 289: 69-74.

[4]Parkes EJ, Duffy BR, Abbott PC. The Jacobi elliptic function method for finding periodic-wave solutions to nonlinear evolution equations. Phys. Lett. A. 2002; 295: 280-286.

[5]Xie Y, Tang J. A unified trial function method in finding the explicit and exact solutions to three NPDEs. Physica Scripta. 2006; 74(2): 197-200. doi: 10.1088/0031-8949/74/2/008

[6]Xie Y. Explicit and exact solutions to the mKdV-SINE-GORDON EQUATION. Modern Physics Letters B. 2008; 22(15): 1471-1485. doi: 10.1142/s0217984908016194

[7]Fu ZT, Liu SD, Liu SK. Notes on Solutions to Burgers-type Equations. Communications in Theoretical Physics. 2004; 41(4): 527-530. doi: 10.1088/0253-6102/41/4/527

[8]Fu Z, Liu S, Liu S. New exact solutions to the KdV–Burgers–Kuramoto equation. Chaos, Solitons & Fractals. 2005; 23(2): 609-616. doi: 10.1016/j.chaos.2004.05.012

[9]Nestor S, Justin M, Douvagai, et al. New Jacobi elliptic solutions and other solutions with quadratic-cubic nonlinearity using two mathematical methods. Asian-European Journal of Mathematics. 2018; 13(02): 2050043. doi: 10.1142/s1793557120500436

[10]Qiu Y, Gao P. New Exact Solutions for the Coupled Nonlinear Schr& #246; dinger Equations with Variable Coefficients. Journal of Applied Mathematics and Physics. 2020; 08(08): 1515-1523. doi: 10.4236/jamp.2020.88117

[11]Nasreen N, Seadawy AR, Lu D. Construction of soliton solutions for modified Kawahara equation arising in shallow water waves using novel techniques. International Journal of Modern Physics B. 2020; 34(07): 2050045. doi: 10.1142/s0217979220500459

[12]Rezazadeh H, Dhawan S, Nestor S, et al. Computational solutions of the generalized Ito equation in nonlinear dispersive systems. International Journal of Modern Physics B. 2021; 35(13): 2150172. doi: 10.1142/s0217979221501721

[13]Kumar S, Kumar D. Analytical soliton solutions to the generalized (3+1)-dimensional shallow water wave equation. Modern Physics Letters B. 2021; 36(02). doi: 10.1142/s0217984921505400

[14]Ishfaq Khan M, Farooq A, Nisar KS, et al. Unveiling new exact solutions of the unstable nonlinear Schrödinger equation using the improved modified Sardar sub-equation method. Results in Physics. 2024; 59: 107593. doi: 10.1016/j.rinp.2024.107593

[15]Mia R, Paul AK. New exact solutions to the generalized shallow water wave equation. Modern Physics Letters B. 2024; 38(31). doi: 10.1142/s0217984924503019

[16]Manafian J, Lakestani M. N-lump and interaction solutions of localized waves to the (2+1)-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation. Journal of Geometry and Physics. 2020; 150: 103598. doi: 10.1016/j.geomphys.2020.103598

[17]Khater MMA. Exploring the dynamics of shallow water waves and nonlinear wave propagation in hyperelastic rods: Analytical insights into the Camassa–Holm equation. Modern Physics Letters B. 2024; 39(08). doi: 10.1142/s0217984924504165

Published

2025-03-21

How to Cite

Xie, Y.-X. (2025). Explicit and accurate solutions for the Benney equation. Journal of AppliedMath, 3(2), 2899. https://doi.org/10.59400/jam2899

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Vol. 3 No. 2 (2025)

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