BÄCKLUND TRANSFORMATIONS AND INFINITE NEW EXPLICIT EXACT SOLUTIONS OF A VARIANT BOUSSINESQ EQUATIONS

Yadong Shang; Huafei Di

doi:10.11948/20230320

2024 Volume 14 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(4): 2140-2157

Next Article Previous Article

Yadong Shang, Huafei Di. BÄCKLUND TRANSFORMATIONS AND INFINITE NEW EXPLICIT EXACT SOLUTIONS OF A VARIANT BOUSSINESQ EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2140-2157. doi: 10.11948/20230320

Citation:

Yadong Shang, Huafei Di. BÄCKLUND TRANSFORMATIONS AND INFINITE NEW EXPLICIT EXACT SOLUTIONS OF A VARIANT BOUSSINESQ EQUATIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 2140-2157. doi: 10.11948/20230320

BÄCKLUND TRANSFORMATIONS AND INFINITE NEW EXPLICIT EXACT SOLUTIONS OF A VARIANT BOUSSINESQ EQUATIONS

Yadong Shang^1,2, ,,
Huafei Di^2,

1.
School of Data Science, Guangzhou Huashang College, Guangzhou, Guangdong 511300, China
2.
School of Mathematics and Information Science, Guangzhou Uinversity, Guangzhou, Guangdong 510006, China

Author Bio: Email address: dihuafei@yeah.net.(H. Di)
Corresponding author: Email address: gzydshang@126.com(Y. Shang)
Fund Project: This work was completed with the support of the Natural Science Foundation of China (11801108), Guangdong Basic and Applied Basic Research Foundation (Nos. 2021A1515010314, 2023A1515030107), the Science and Technology Planning Project Guangzhou City (No. 202201010111)

Abstract

Abstract

This paper deals with a variant Boussinesq equations which describes the propagation of shallow water waves in a lake or near an ocean beach. We derive out two hetero-Bäcklund transformations between the variant Boussinesq equations and two linear parabolic equations by using the extended homogeneous balance method. We also obtain two hetero-Bäcklund transformations between the variant Boussinesq equations and Burgers equations. Furthermore, we obtain two hetero-Bäcklund transformation between the variant Boussinesq equations and heat equations. By using these Bäcklund transformations and so-called "seed solution", we obtain a large number of explicit exact solutions of the variant Boussinesq equations. Especially, The infinite explicit exact singular wave solutions of variant Boussinesq equations are obtained for the first time. It is worth noting that these singular wave solutions of variant Boussinesq equations will blow up on some lines or curves in the (x, t) plane. These facts reflect the complexity of the structure of the solution of variant Boussinesq equations. It also reflects the complexity of shallow water wave propagation from one aspect.
- A variant Boussinesq equations /
- nonlinear transformation /
- exact linearization /
- explicit exact solution /
- singular wave solution
MSC: 35C07, 37G10

FullText(HTML)

References

[1]	M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, 1991. Google Scholar
[2]	L. Akinyemia, M. Senolb and O. S. Iyiola, Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method, Math. Comput. Simula., 2021, 182, 211–233. doi: 10.1016/j.matcom.2020.10.017 CrossRef Google Scholar
[3]	G. Arindam and M. Sarit, The first integral method and some nonlinear models, Comput. Appl. Math., 2021, 40, 79, 16 pp. Google Scholar
[4]	D. J. Arrigo, P. Broadbridge and J. M. Hill, Non-classical symmetry solutions and the methods of Bluman-Cole and Clarkson-Kruskal, J. Math. Phys., 1993, 34, 4692–4703. doi: 10.1063/1.530365 CrossRef Google Scholar
[5]	B. Bibekananda, M. Hemanta and Z. D. Dia, Exact solution of the time fractional variant Boussinesq-Burgers equations, Appl. Math., 2021, 66(3), 437–449. doi: 10.21136/AM.2021.0269-19 CrossRef Google Scholar
[6]	L. J. F. Broer, Approximate equations for long water waves, Appl. Sci. Res., 1975, 31(5), 377–395. doi: 10.1007/BF00418048 CrossRef Google Scholar
[7]	D. Evgueni, Travelling waves in the Boussinesq type systems, J. Math. Pures Appl., 2022, 163(9), 1–10. Google Scholar
[8]	E. G. Fan, Two new applications of the homogeneous balance method, Phys. Lett. A, 2000, 265, 353–357. doi: 10.1016/S0375-9601(00)00010-4 CrossRef Google Scholar
[9]	E. G. Fan and Y. C. Hon, A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves, Chaos, Solitons and Fractals., 2003, 15, 559–566. doi: 10.1016/S0960-0779(02)00144-3 CrossRef Google Scholar
[10]	E. G. Fan and H. Q. Zhang, A new approach to Bäcklund transformations of nonlinear evolution equations, Appl. Math. Mech., 1998, 19(7), 645–650. doi: 10.1007/BF02452372 CrossRef Google Scholar
[11]	E. G. Fan and H. Q. Zhang, Bäcklund transformation and exact solutions for Whitham-Broer-Kaup equations in shallow water, Appl. Math. Mech., 1998, 19(8), 713–716. doi: 10.1007/BF02457745 CrossRef Google Scholar
[12]	E. G. Fan and H. Q. Zhang, Symmetry reductions and similarity solutions of variant Boussinesq equation systems, (Chinese) Acta. Math. Sci., 1999, 19(4), 373–378. Google Scholar
[13]	B. L. Guo, Nonlinear Evolution Equations, Shanghai Scientific and Technological Education Publishing House, Shanghai, 1995. Google Scholar
[14]	H. Guo, T. Z. Xu, S. J. Yang and G. W. Wang, Aanlytical study of solitons for the variant Boussinesq equations, Nonliinear Dyn., 2017, 88, 1139–1146. doi: 10.1007/s11071-016-3300-5 CrossRef Google Scholar
[15]	S. Hood, New exact solutions of Burgers's equation-an extension to the directmethod of Clarkson and Kruskal, J. Math. Phys., 1995, 36, 1971–1990. doi: 10.1063/1.531097 CrossRef Google Scholar
[16]	D. J. Kaup, A higher-order water wave equation and the method for solving it, Prog. Theor. Phys., 1975, 54(2), 396–408. doi: 10.1143/PTP.54.396 CrossRef Google Scholar
[17]	B. A. Kupershmidt, Mathematics of dispersive water waves, Commun. Math. Phys., 1985, 99, 51–73. doi: 10.1007/BF01466593 CrossRef Google Scholar
[18]	W. H. Liu and Z. F. Zhang, Lie symmetry analysis, analytical solutions and conservation laws to the coupled time fractional variant Boussinesq equations, Waves Random Complex Media, 2021, 31(1), 182–197. doi: 10.1080/17455030.2019.1577583 CrossRef Google Scholar
[19]	R. Naz, F. Mahomed and T. Hayat, Conservation laws for the third-order variant Boussinesq system, App. Math. Lett., 2010, 23(8), 8830886. Google Scholar
[20]	J. E. Okeke, R. Narain and K. S. Govinder, New exact solutions of a generalised Boussinesq equation with damping term and a system of variant Boussinesq equations via double reduction theory, J. Appl. Anal. Comp., 2018, 8(2), 471–485. Google Scholar
[21]	R. L. Sachs, On the integrable variant of the Boussinesq system: Painlevé property, rational solutions, a related many-body system, and equivalence with the AKNS hierarchy, Phys. D., 1988, 30(1–2), 1–27. Google Scholar
[22]	M. Sarit, G. Arindam and R. C. Asesh, Exact solutions and symmetry analysis of a new equation invariant under scaling of dependent variable, Phys. Scr., 2019, 94(8), 085212, 7 pp. Google Scholar
[23]	M. Sarit, G. Arindam and R. C. Asesh, Exact solutions and symmetry snalysis of a Boussinesq type equation for longitudinal waves through a magneto-electro-elastic circular rod, Int. J. Appl. Comput. Math., 2021, 7, 171, 14 pp. Google Scholar
[24]	M. L. Wang, Solitray wave solutions for variant Boussinesq equations, Phys. Lett. A, 1995, 199, 169–172. doi: 10.1016/0375-9601(95)00092-H CrossRef Google Scholar
[25]	Z. Y. Yan and H. Q. Zhang, New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics, Phys. Lett. A, 1999, 252(6), 291–296. doi: 10.1016/S0375-9601(98)00956-6 CrossRef Google Scholar
[26]	Z. Y. Yan and H. Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water, Phys. Lett. A, 2001, 285, 355–362. doi: 10.1016/S0375-9601(01)00376-0 CrossRef Google Scholar
[27]	X. F. Yang, Z. C. Deng, Q. J. Li and Y. Wei, Exact combined traveling wave solutions and multi-sympletic structure of the varianr Boussinesq Whitham-Broer-Kaup type equations, Commun. Nonlin. Sci. Simul., 2016, 36, 1–13. doi: 10.1016/j.cnsns.2015.11.015 CrossRef Google Scholar
[28]	E. Yomba, The extended Fan's sub-equation method and its application to KdV-MKdV, BKK, and variant Boussinesq equations, Phys. Lett. A, 2005, 336, 463–476. doi: 10.1016/j.physleta.2005.01.027 CrossRef Google Scholar
[29]	Y. B. Yuan, D. M. Pu and S. M. Li, Bifurcations of travelling wave solutions in variant Boussinesq equations, Appl. Math. Mech., 2006, 27(7), 811–822. Google Scholar
[30]	J. F. Zhang, Multi-solitary wave solutions for variant Boussinesq equations and Kupershmidt equations, Appl. Math. Mech., 2000, 21(2), 193–198. Google Scholar
[31]	P. Zhang, New exact solutions to breaking soliton equations and Whitham-Broer-Kaup equations, Appl. Math. Comput., 2010, 217, 1688–1696. Google Scholar
[32]	W. G. Zhang, Q. Liu, X. Li and B. L. Guo, Shape analysis of bounded traveling wave solutions and solution to the generalized Whitham-Broer-Kaup equation with dissipation terms, Chin. Ann. Math., 2012, 33B(2), 281–308. Google Scholar
[33]	W. G. Zhang, Q. Liu, Z. M. Li and X. Li, Bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect, Acta Math. Sci., 2014, 34B(3), 941–959. Google Scholar