Extended Jacobi elliptic function solutions for general boussinesq systems
DOI:
https://doi.org/10.31349/RevMexFis.69.021401Keywords:
Jacobi elliptic function method, travelling wave solutions, boussinesq systemAbstract
In this research paper, we have utilized the Jacobi elliptic function expansion method to obtain the exact solutions of (1+1)- dimensional Boussinesq System (GBQS). The most important difference that distinguishes this method from other methods is the parameters included in the auxiliary equation F’ (ξ) = Ö P F4(ξ) + QF2(ξ) + R. As far as the authors know, there is no other study in which such a variety of solutions has been given. Depending on P, Q and R, nineteen the solitary wave and periodic wave solutions are obtained at their limit conditions. In addition, 3D and contour plot graphics for the constructed waves are investigated with the computer package program by giving special values to the parameters involved. The validity and reliability of the method is examined by its applications on a class of nonlinear evolution equations of special interest in nonlinear mathematical physics. The results were acquired to verify that the recommended method is applicable and reliable for the analytic treatment of a wide application of nonlinear phenomena
References
Zhang J E, Chen C L, Li Y S. On Boussinesq models of constant depth. Phys Fluids, 16 (2004) 1287, https://doi.org/10.1063/1.1688323.
Chen C L, Lou S Y, Li Y S. Solitary wave solutions for a general Boussinesq type fluid model. CommuniNonlinear Sci Numer Simul, 9 (2004) 583, https://doi.org/10.1016/S1007-5704(03)00009-1
J.L. Bona, M. Chen, J.-C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: The nonlinear theory. Nonlinearity 17 (2004) 925, https://doi.org/10.1088/0951-7715/17/3/010
D.J. Kaup. A higher-order wave equation and the method for solving it. Prog. Theor. Phys. 54 (1975) 396, https://doi.org/10.1143/PTP.54.396
Tchier, Fairouz, et al., Time fractional third-order variant Boussinesq system: Symmetry analysis, explicit solutions, conservation laws and numerical approximations. The European Physical Journal Plus 133 (2018) 240, https://doi.org/10.1140/epjp/i2019-12518-1
Meng, De-Xin et al., Solitonic solutions for a variablecoefficient variant Boussinesq system in the long gravity waves. Applied Mathematics and Computation 215 (2009) 1744, https://doi.org/10.1016/j.amc.2009.07.039
Yas¸ar, E., and I. B. Giresunlu, Lie symmetry reductions, exact solutions and conservation laws of the third order variant Boussinesq system. Acta Physica Polonica A 128 (2015) 252, https://doi.org/10.12693/APhysPolA.128.252
Zhang, Jin E., Chunli Chen, and Yishen Li. On Boussinesq models of constant depth. Physics of Fluids 16 (2004) 1287, https://doi.org/10.1063/1.1688323
Li, JiBin, Bifurcations of travelling wave solutions for two generalized Boussinesq systems. Science in China Series A: Mathematics 51 (2008) 1577, https://doi.org/10.1007/s11425-008-0038-7
Shou-Jun, Huang and Chen Chun-Li. Study on solitary waves of a general Boussinesq model. Communications in Theoretical Physics 48 (2007) 773, https://doi.org/10.1088/0253-6102/48/5/002
Chen, Chunli, Shoujun Huang, and Jin E. Zhang. On head-on collisions between two solitary waves of Nwogu’s Boussinesq model. Journal of the Physical Society of Japan 77 (2008) 014003, https://doi.org/10.1143/JPSJ.77.014003
San, Sait, Invariant analysis of nonlinear time fractional Qiao equation. Nonlinear Dynamics 85 (2016) 2127, https://doi.org/10.1007/s11071-016-2818-x
Yaşar, Emrullah, Sait San, and Yeşim Sağlam Özkan, Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation. Open Physics 14 (2016) 37, https://doi.org/10.1515/phys-2016-0007
San, Sait, and Emrullah Yas¸ar, On the conservation laws of Derrida-Lebowitz-Speer-Spohn equation, Communications in Nonlinear Science and Numerical Simulation 22 (2015) 1297, https://doi.org/10.1016/j.cnsns.2014.08.031
T. Y. Wu, Long waves in ocean and coastal waves, J. Eng. Mech. 107 (1981) 501, https://doi.org/10.1061/JMCEA3.0002722
O. Nwogu, An alternative form of the Boussinesq equations for nearshorewave propagation, J. Waterw., Port, Coastal, Ocean Eng. 119 (1993) 618, https://doi.org/10.1061/(ASCE)0733-950X(1993)119:6(618)
Tang, Bo, et al. A generalized fractional sub-equation method for fractional differential equations with variable coefficients. Physics Letters A 376 (2012) 2588, https://doi.org/10.1016/j.physleta.2012.07.018
H. Bulut, and P. Yusuf, Modified trial equation method to the nonlinear fractional Sharma-Tasso-Olever equation. International Journal of Modeling and Optimization 3 (2013) 353, https://doi.org/10.7763/IJMO.2013.V3.297
N. Taghizadeh, et al. Application of the simplest equation method to some time-fractional partial differential equations. Ain Shams Engineering Journal 4 (2013) 897, https://doi.org/10.1016/j.asej.2013.01.006
S. Tuluce Demiray, P. Yusuf, and B. Hasan, Generalized Kudryashov method for time-fractional differential equations. Abstract and applied analysis. 2014 (2014) 901540, https://doi.org/10.1155/2014/901540
H. Bulut, P. Yusuf and S. Tuluce Demiray, Exact solutions of time-fractional KdV equations by using generalized Kudryashov method. International Journal of Modeling and Optimization 4 (2014) 315, https://doi.org/10.7763/IJMO.2014.V4.392
E. Yas¸ar, Y. Yıldırım, and C. Masood Khalique, Lie symmetry analysis, conservation laws and exact solutions of the seventhorder time fractional Sawada–Kotera–Ito equation. Results in physics 6 (2016) 322, https://doi.org/10.1016/j.rinp.2016.06.003
Z. Fu, et al. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Physics Letters A 290 (2001) 72, https://doi.org/10.1016/S0375-9601(01)00644-2
Z. Yan, Abundant families of Jacobi elliptic function solutions of the (2+ 1)-dimensional integrable Davey–Stewartsontype equation via a new method. Chaos, Solitons & Fractals 18 (2003) 299, https://doi.org/10.1016/S0960-0779(02)00653-7
D. Lü, Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos, Solitons & Fractals 24 (2005) 1373, https://doi.org/10.1016/j.chaos.2004.09.085
E. Tala-Tebue, and E. M. E. Zayed, New Jacobi elliptic function solutions, solitons and other solutions for the (2+ 1)- dimensional nonlinear electrical transmission line equation. The European Physical Journal Plus 133 (2018) 314, https://doi.org/10.1140/epjp/i2018-12118-7
R. D. Pankaj, B. Singh, and A. Kumar, New extended Jacobi elliptic function expansion scheme for wave-wave interaction in ionic media. Nanosystems: Physics, Chemistry, Mathematics 9 (2018) 581, https://doi.org/10.17586/2220-8054-2018-9-5-581-585
A. Ebaid, and E. H. Aly, Exact solutions for the transformed reduced Ostrovsky equation via the F-expansion method in terms of Weierstrass-elliptic and Jacobian-elliptic functions. Wave Motion 49 (2012) 296, https://doi.org/10.1016/j.wavemoti.2011.11.003
R. D. Pankaj, A. Kumar and C. Lal, A novel narration for two long waves interaction through the elaboration scheme, J. Inter. Maths., 25 (2022) 79, https://doi.org/10.1080/09720502.2021.2010768.
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