Soliton Solutions of Variant Boussinesq Equations Through Exp-function Method
Keywords:
Soliton solutions, Exp-function technique, Variant Boussinesq equations, Maple 18Abstract
This research article presents soliton solutions of
Variant Boussinesq equations. The Boussinesq equation governs
the dynamics of shallow water waves that are seen in various
places like sea beaches, lakes and rivers. By a suitable
transformation the nonlinear partial differential equation is
converted into nonlinear ordinary differential equation. The expfunction method is applied to solve the mathematical problem.
The novel type results based on the solitary wave structures
contributes a lot in the regime of nonlinear wave phenomena. It is
observed that scheme is highly trustworthy and may be extended
to other nonlinear models represented in the form of highly
nonlinear differential equations.
References
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Applied Mathematics E-Notes. Vol. 8, pp. 58-66, 2008.
[8] Y. B. Zhou, M.L. Wang and Y. M. Wang. Periodic wave solutions to
coupled KdV equations with variable coefficients. Physics Letters A,
Vol. 308, pp. 31–36, 2003.
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for new exact solutions of the nonlinear partial differential equations.
International Journal of the Physical Sciences. Vol. 6(29), pp. 6706-
6716, 2011.
[10] N. Taghizadeh and M. Mirzazadeh. The first integral method to some complex nonlinear partial differential equations. Journal of Computational and Applied Mathematics, Vol. 235, pp. 4871-4877,
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[11] W. X. Ma, J. H. Lee, A transformed rational function method and exact
solutions to the (3+1) dimensional Jimbo Miwa equation. Chaos,
Solitons and Fractals, Vol. 42, pp. 1356-1363, 2009.
[12] W.X. Ma, T. Huang, Y. Zhang, A multiple exp-function method for
nonlinear differential equations and its application. Physica Scripta, Vol.
82, 2010.
[13] W. X. Ma, M. Chen. Direct search for exact solutions to the nonlinear
Schrödinger equation. Applied Mathematics and Computation, Vol. 215,
pp. 2835-2842, 2009.
[14] W.X. Ma. Generalized bilinear differential equations. Studies in
Nonlinear Sciences, Vol. 2, pp. 140-144, 2011.
[15] N. A. Kudryashov. On types of nonlinear non-integrable equations with
exact solutions. Physics. Letters A, Vol. 155 pp.269-275, 1991.
[16] N. A. Kudryashov. Exact solutions of the generalized Kuramoto-
Sivashinsky equation. Physics Letters A, Vol. 147 pp. 287-291, 1990.
[17] Y. Chen and Q. Wang. Extended Jacobi elliptic function rational
expansion method and abundant families of Jacobi elliptic functions
solutions to (1+1)-dimensional dispersive long wave equation. Chaos,
Solitons and Fractals. Vol. 24, pp. 745-757, 2005.
[18] E. Yusufoglu and A. Bekir, Exact solution of coupled nonlinear
evolution equations. Chaos, Solitons and Fractals, Vol. 37. pp. 842-848,
2008..
[19] A. T. Ali, New generalized Jacobi elliptic function rational expansion
method. Journal of Computational and Applied Mathematics, Vol. 235,
pp. 4117-4127, 2011.
[20] Z. Wang and H. Q. Zhang, A new generalized Riccati equation rational
expansion method to a class of nonlinear evolution equation with
nonlinear terms of any order. Applied Mathematics and Computation,
Vol. 186, pp. 693-704, 2007.
[21] M. Wang and X. Li. Applications of f-expansion toperiodic wave
solutions for a new amiltonian amplitude equation. Chaos, Solitons and
Fractals. Vol. 24, pp. 1257- 1268, 2005.
[22] M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution
Equations and Inverse Scattering Transform. Cambridge University
Press, Cambridge, 1991.
[23] R. Hirota. Exact solution of the KdV equation for multiple collisions of
solutions. Physical Review Letters, Vol. 27, pp. 1192-1194, 1971.
[24] J.H. He and X.H. Wu. Exp-function method for nonlinear wave
equations. Chaos, Solitons and Fractals, Vol. 30 (3), pp. 700–708, 2006.
[25] A. Bekir, A. Boz. Exact solutions for nonlinear evolution equation using
Exp-function method. Physics Letters A, Vol. 372, pp. 1619–1625,
2008.
[26] J. H. He. An elementary introduction of recently developed asymptotic
methods and nano-mechanics in textile engineering. International
Journal of Modern Physics B Vol. 22 (21), pp. 3487-4578, 2008.
[27] J. H. He and M. A. Abdou, New periodic solutions for nonlinear
evolution equation using exp-method. Chaos, Solitons and Fractals, Vol.
34, pp. 1421-1429, 2007.
[28] S. T. Mohyud-Din, M. A. Noor, K. I. Noor and A. Waheed. Expfunction
method for generalized travelling solutions of Lax equations.
International Journal of Modern Physics B. 2009.
[29] X. H.Wu and J. H. He, Solitary solutions, periodic solutions and
compacton like solutions using the Exp-function method. Computers and
Mathematics with Applications, Vol. 54, pp. 966-986, 2007.
[30] M.A. Mostafa Khater and H. M.Emad Zahran. Modified extended tanh
function method and its applications to the Bogoyavlenskii equation.
Applied Mathematical Modelling. Vol. 40, pp. 1769-1775, 2016.
[31] Mostafa M.A. Khater. Exact traveling wave solutions for the generalized
Hirota-Satsuma couple KdV system using the exp (-phi)-expansion
method. Cogent Mathematics. Vol. 3(1), pp. 1-16, 2016.
[32] A. E. Mahmoud Abdelrahman and M.A. Mostafa Khater. Exact
traveling wave solutions for Fitzhugh-Nagumo (FN) equation and
modified Liouville equation, International Journal of Computer
Applications, Vol. 113(3), pp. 1-7, 2015.
[33] H. M Emad Zahran and M. A. Mostafa Khater. Exact traveling wave
solutions for nonlinear dynamics of microtubles -A New Model and The Kundu- Eckhaus Equation. Asian Journal of Mathematics and Computer
Research, Vol. 4(4), pp. 195- 206, 2015.
[34] M.A Mostafa. Khater, H.M. Emad Zahran and S.M. Maha Shehata.
Solitary wave solution of the generalized Hirota-Satsuma coupled KdV
system. Journal of the Egyptian Mathematical Society, Vol. 25(1), pp. 8-
12, 2016.
[35] D. S. Wang, S. Y. Tian, Y. Liu. Integrability and bright soliton solutions
to the coupled nonlinear Schrödinger equation with higher-order effects.
Applied Mathematics Computation, Vol. 229, pp. 296-309, 2014.
[36] D. S. Wang, X. Wei. Integrability and exact solutions of a twocomponent
Korteweg–de Vries system. Applied Mathematics Letters,
Vol. 51, pp. 60-67, 2016.
[37] D. S. Wang, D. J. Zhang, J. Yang. Integrable properties of the general
coupled nonlinear Schrodinger equations, Journal of Mathematical
Physics. Vol. 51, 2010.
[38] W.X. Ma, B. Fuchssteiner. Explicit and exact solutions to a Kolmogorov-
Petrovskii-Piskunov equation, International Journal of Mechanics,
Vol. 31, pp. 329-338, 1996.
[39] W. X. Ma. A refined invariant subspace method and applications to
evolution equations. Science China Mathematics, Vol. 55, pp. 1778-
1796, 2012.
[40] W.X. Ma, Lump solutions to the Kadomtsev–Petviashvili equation.
Physics Letters A, Vol. 379, pp. 1975–1978, 2015.
British Association for Advancement of Sciences, York, September
1844 (London 1845), pp. 311-390.
[2] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura. Method
for Solving the korteweg-deVries equation. Physics Revision Letters,
Vol. 19 pp. 1095-1099, 1967.
[3] C. Rogers and W. F. Shadwick, Backlund Transformations. Academic
Press, New York, 1982.
[4] S. T. Mohyud-Din. Variational iteration method for solving Fifth-order
boundary value problems using He’s polynomials. 2008.
[5] M. A. Abdou, The extended tanh-method and its applications for solving
nonlinear physical models. Applied Mathematics and Computation,
Vol.190, pp. 988-996, 2007.
[6] S. A. El-Wakil, M. A. Abdou, E. K. El-Shewy and A. Hendi. (G?/ G)
expansion method equivalent to the extended tanh-function method.
Physics Script. Vol. 81 pp. 35011- 35014, 2010.
[7] S. Zhang and T. C. Xia. A further improved tanh-function method
exactly solving the (2+1)-dimensional dispersive long wave equations.
Applied Mathematics E-Notes. Vol. 8, pp. 58-66, 2008.
[8] Y. B. Zhou, M.L. Wang and Y. M. Wang. Periodic wave solutions to
coupled KdV equations with variable coefficients. Physics Letters A,
Vol. 308, pp. 31–36, 2003.
[9] H. Naher, F. A. Abdullah and M. A. Akbar. The exp-function method
for new exact solutions of the nonlinear partial differential equations.
International Journal of the Physical Sciences. Vol. 6(29), pp. 6706-
6716, 2011.
[10] N. Taghizadeh and M. Mirzazadeh. The first integral method to some complex nonlinear partial differential equations. Journal of Computational and Applied Mathematics, Vol. 235, pp. 4871-4877,
2011.
[11] W. X. Ma, J. H. Lee, A transformed rational function method and exact
solutions to the (3+1) dimensional Jimbo Miwa equation. Chaos,
Solitons and Fractals, Vol. 42, pp. 1356-1363, 2009.
[12] W.X. Ma, T. Huang, Y. Zhang, A multiple exp-function method for
nonlinear differential equations and its application. Physica Scripta, Vol.
82, 2010.
[13] W. X. Ma, M. Chen. Direct search for exact solutions to the nonlinear
Schrödinger equation. Applied Mathematics and Computation, Vol. 215,
pp. 2835-2842, 2009.
[14] W.X. Ma. Generalized bilinear differential equations. Studies in
Nonlinear Sciences, Vol. 2, pp. 140-144, 2011.
[15] N. A. Kudryashov. On types of nonlinear non-integrable equations with
exact solutions. Physics. Letters A, Vol. 155 pp.269-275, 1991.
[16] N. A. Kudryashov. Exact solutions of the generalized Kuramoto-
Sivashinsky equation. Physics Letters A, Vol. 147 pp. 287-291, 1990.
[17] Y. Chen and Q. Wang. Extended Jacobi elliptic function rational
expansion method and abundant families of Jacobi elliptic functions
solutions to (1+1)-dimensional dispersive long wave equation. Chaos,
Solitons and Fractals. Vol. 24, pp. 745-757, 2005.
[18] E. Yusufoglu and A. Bekir, Exact solution of coupled nonlinear
evolution equations. Chaos, Solitons and Fractals, Vol. 37. pp. 842-848,
2008..
[19] A. T. Ali, New generalized Jacobi elliptic function rational expansion
method. Journal of Computational and Applied Mathematics, Vol. 235,
pp. 4117-4127, 2011.
[20] Z. Wang and H. Q. Zhang, A new generalized Riccati equation rational
expansion method to a class of nonlinear evolution equation with
nonlinear terms of any order. Applied Mathematics and Computation,
Vol. 186, pp. 693-704, 2007.
[21] M. Wang and X. Li. Applications of f-expansion toperiodic wave
solutions for a new amiltonian amplitude equation. Chaos, Solitons and
Fractals. Vol. 24, pp. 1257- 1268, 2005.
[22] M. J. Ablowitz and P. A. Clarkson. Solitons, Nonlinear Evolution
Equations and Inverse Scattering Transform. Cambridge University
Press, Cambridge, 1991.
[23] R. Hirota. Exact solution of the KdV equation for multiple collisions of
solutions. Physical Review Letters, Vol. 27, pp. 1192-1194, 1971.
[24] J.H. He and X.H. Wu. Exp-function method for nonlinear wave
equations. Chaos, Solitons and Fractals, Vol. 30 (3), pp. 700–708, 2006.
[25] A. Bekir, A. Boz. Exact solutions for nonlinear evolution equation using
Exp-function method. Physics Letters A, Vol. 372, pp. 1619–1625,
2008.
[26] J. H. He. An elementary introduction of recently developed asymptotic
methods and nano-mechanics in textile engineering. International
Journal of Modern Physics B Vol. 22 (21), pp. 3487-4578, 2008.
[27] J. H. He and M. A. Abdou, New periodic solutions for nonlinear
evolution equation using exp-method. Chaos, Solitons and Fractals, Vol.
34, pp. 1421-1429, 2007.
[28] S. T. Mohyud-Din, M. A. Noor, K. I. Noor and A. Waheed. Expfunction
method for generalized travelling solutions of Lax equations.
International Journal of Modern Physics B. 2009.
[29] X. H.Wu and J. H. He, Solitary solutions, periodic solutions and
compacton like solutions using the Exp-function method. Computers and
Mathematics with Applications, Vol. 54, pp. 966-986, 2007.
[30] M.A. Mostafa Khater and H. M.Emad Zahran. Modified extended tanh
function method and its applications to the Bogoyavlenskii equation.
Applied Mathematical Modelling. Vol. 40, pp. 1769-1775, 2016.
[31] Mostafa M.A. Khater. Exact traveling wave solutions for the generalized
Hirota-Satsuma couple KdV system using the exp (-phi)-expansion
method. Cogent Mathematics. Vol. 3(1), pp. 1-16, 2016.
[32] A. E. Mahmoud Abdelrahman and M.A. Mostafa Khater. Exact
traveling wave solutions for Fitzhugh-Nagumo (FN) equation and
modified Liouville equation, International Journal of Computer
Applications, Vol. 113(3), pp. 1-7, 2015.
[33] H. M Emad Zahran and M. A. Mostafa Khater. Exact traveling wave
solutions for nonlinear dynamics of microtubles -A New Model and The Kundu- Eckhaus Equation. Asian Journal of Mathematics and Computer
Research, Vol. 4(4), pp. 195- 206, 2015.
[34] M.A Mostafa. Khater, H.M. Emad Zahran and S.M. Maha Shehata.
Solitary wave solution of the generalized Hirota-Satsuma coupled KdV
system. Journal of the Egyptian Mathematical Society, Vol. 25(1), pp. 8-
12, 2016.
[35] D. S. Wang, S. Y. Tian, Y. Liu. Integrability and bright soliton solutions
to the coupled nonlinear Schrödinger equation with higher-order effects.
Applied Mathematics Computation, Vol. 229, pp. 296-309, 2014.
[36] D. S. Wang, X. Wei. Integrability and exact solutions of a twocomponent
Korteweg–de Vries system. Applied Mathematics Letters,
Vol. 51, pp. 60-67, 2016.
[37] D. S. Wang, D. J. Zhang, J. Yang. Integrable properties of the general
coupled nonlinear Schrodinger equations, Journal of Mathematical
Physics. Vol. 51, 2010.
[38] W.X. Ma, B. Fuchssteiner. Explicit and exact solutions to a Kolmogorov-
Petrovskii-Piskunov equation, International Journal of Mechanics,
Vol. 31, pp. 329-338, 1996.
[39] W. X. Ma. A refined invariant subspace method and applications to
evolution equations. Science China Mathematics, Vol. 55, pp. 1778-
1796, 2012.
[40] W.X. Ma, Lump solutions to the Kadomtsev–Petviashvili equation.
Physics Letters A, Vol. 379, pp. 1975–1978, 2015.
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Published
2017-12-01
How to Cite
Dr. Qazi Mahmood Ul-Hassan. (2017). Soliton Solutions of Variant Boussinesq Equations Through Exp-function Method. University of Wah Journal of Science and Technology (UWJST), 1(1), 24–30. Retrieved from https://uwjst.org.pk/index.php/uwjst/article/view/6
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