American Institute of Mathematical Sciences

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December  2015, 8(6): 1155-1164. doi: 10.3934/dcdss.2015.8.1155

Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation

Anwar Ja'afar Mohamad Jawad 1, , Mohammad Mirzazadeh 2, and  Anjan Biswas 3,

1. 

Computer Engineering Technique Department Al-Rafidain, University College, Baghdad, Iraq
2. 

Department of Engineering Sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, P.C. 44891-63157, Rudsar-Vajargah, Iran
3. 

Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, United States

Received  May 2015 Revised  August 2015 Published  December 2015

This paper obtains soliton and other solutions to the Gardner-Kadomtsev-Petviashvili equation that models shallow water wave equation in (1+2)-dimensions. There are three types of integration architectures that will be employed in order to obtain several forms of solution to this model. These are traveling wave hypothesis, improved $G^{\prime}/G$-expansion method and finally the tanh-coth hypothesis. The constraint conditions that are needed, for these solutions to exist, are also reported.
Keywords: traveling waves, tanh-coth method, Gardner-KP equation., $G^{\prime}/G$-expansion.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Anwar Ja'afar Mohamad Jawad, Mohammad Mirzazadeh, Anjan Biswas. Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1155-1164. doi: 10.3934/dcdss.2015.8.1155
References:
[1]

M. Antonova and A. Biswas, Adiabatic parameter dynamics of perturbed solitons, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 734-748. doi: 10.1016/j.cnsns.2007.12.004.  Google Scholar

[2]

A. H. Bhrawy, M. A. Abdelkawy, S. Kumar and A. Biswas, Solitons and other solutions to Kadomtsev-Petviashvili equation of B-type, Romanian Journal of Physics, 58 (2013), 729-748.  Google Scholar

[3]

A. Biswas and E. Zerrad, Soliton perturbation theory for the Gardner equation, Advanced Studies in Theoretical Physics, 2 (2008), 787-794.  Google Scholar

[4]

A. Biswas and A. Ranasinghe, 1-soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 214 (2009), 645-647. doi: 10.1016/j.amc.2009.04.001.  Google Scholar

[5]

A. Biswas and A. Ranasinghe, Topological 1-soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 217 (2010), 1771-1773. doi: 10.1016/j.amc.2009.09.042.  Google Scholar

[6]

R. Choudhury and S. K. Das, Viscelastic MHD free convective flow through porous media in presence of radiation and chemical reaction with heat and mass transfer, Journal of Applied Fluid Mechanics, 7 (2014), 603-609. Google Scholar

[7]

G. Ebadi, N. Y. Fard, A. H. Bhrawy, S. Kumar, H. Triki, A. Yildirim and A. Biswas, Solitons and other solutions to (3+1)-dimensional extended Kadomtsev-Petviashvili equation with power law nonlinearity, Romaninan Reports in Physics, 65 (2013), 27-62. Google Scholar

[8]

M. Eslami, M. Mirzazadeh and A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger's equation in optical fibers with time-dependent coefficients by simplest equation approach, Journal of Modern Optics, 60 (2013), 1627-1636. doi: 10.1080/09500340.2013.850777.  Google Scholar

[9]

M. Eslami and M. Mirzazadeh, Topological 1-soliton solution of nonlinear Schrodinger equation with dual-power law nonlinearity in nonlinear optical fibers, European Physical Journal, Plus, 128 (2013), p140. Google Scholar

[10]

E. V. Krishnan, H. Triki, M. Labidi and A. Biswas, A study of shallow water waves with Gardner's equation, Nonlinear Dynamics, 66 (2011), 497-507. doi: 10.1007/s11071-010-9928-7.  Google Scholar

[11]

S. Kundu and K. Ghoshal, An explicit model for concentration distribution using biquadratic log-wake law in an open channel flow, Journal of Applied Fluid Mechanics, 6 (2013), 339-350. Google Scholar

[12]

Z. G. Makukula and S. S. Motsa, Spectral homotopy analysis method for PDEs that model the unsteady Von Karma swirling flow, Journal of Applied Fluid Mechanics, 7 (2014), 711-718. Google Scholar

[13]

M. Mirzazadeh, M. Eslami and A. Biswas, Soliton solutions of the generalized Klein-Gordon equation by using ${G'}/G$-expansion method, Computational and Applied Mathematics, 33 (2014), 831-839. doi: 10.1007/s40314-013-0098-3.  Google Scholar

[14]

M. Mirzazadeh and M. Eslami, Exact solutions for nonlinear variants of Kadomtsev-Petviashvili (n, n) equation using functional variable method, Pramana, 81 (2013), 225-236. Google Scholar

[15]

A. Nazarzadeh, M. Eslami and M. Mirzazadeh, Exact solutions of some nonlinear partial differential equations using functional variable method, Pramana, 81 (2013), 225-236. doi: 10.1007/s12043-013-0565-9.  Google Scholar

[16]

D. Pal and S. Chatterjee, Effects of radiation on Darcey-Forchheimer convective flow over a stretching sheet in a micropolar fluid with a non-uniform heat source/sink, Journal of Applied Fluid Mechanics, 8 (2015), 207-212. Google Scholar

[17]

P. Ram and V. Kumar, Rotationally symmetric ferrofluid flow and heat transfer in porous medium with variable viscosity and viscous dissipation, Journal of Applied Fluid Mechanics, 7 (2014), 357-366. Google Scholar

[18]

S. M. Shafiof, Z. Bagheri and Sousaraei, New solutions for positive and negative Gardner-KP equations, World Applied Science Journal, 13 (2011), 662-666. Google Scholar

[19]

N. Taghizadeh and M. Mirzazadeh, The simplest equation method to study perturbed nonlinear Schrodinger's equation with Kerr law nonlinearity, Communications in Nonlinear Science and Numerical Simulations, 17 (2012), 1493-1499. doi: 10.1016/j.cnsns.2011.09.023.  Google Scholar

[20]

N. Taghizadeh, M. Mirzazadeh and F. Farahrooz, Exact soliton solutions of the modified KdV-KP equation and the Burgers-KP equation by using the first integral method, Applied Mathematical Modelling, 35 (2011), 3991-3997. doi: 10.1016/j.apm.2011.02.001.  Google Scholar

[21]

N. Taghizadeh, M. Mirzazadeh and A. Samiei Paghaleh, Exact solutions of some nonlinear evolution equations via the first integral method, Ain Shams Engineering Journal, 4 (2013), 493-499. doi: 10.1016/j.asej.2012.10.002.  Google Scholar

[22]

W. M. Taha, M. S. M. Noorani and I. Hashim, New exact solutions of sixth-order thin-film equation, Journal of King Saud University- Science, 26 (2014), 75-78. doi: 10.1016/j.jksus.2013.07.001.  Google Scholar

[23]

F. Tascan, A. Bekir and M. Koparan, Travelling wave solutions of nonlinear evolutions by using the first integral method, Communications in Nonlinear Science and Numerical Simulations, 14 (2009), 1810-1815. doi: 10.1016/j.cnsns.2008.07.009.  Google Scholar

[24]

F. Tascan and A. Bekir, Travelling wave solutions of the Cahn-Allen equation by using first integral method, Applied Mathematics and Computation, 207 (2009), 279-282. doi: 10.1016/j.amc.2008.10.031.  Google Scholar

[25]

H. Triki, B. J. M. Sturdevant, T. Hayat, O. M. Aldossary and A. Biswas, Shock wave solutions of the variants of Kadomtsev-Petviashvili equation, Canadian Journal of Physics, 89 (2011), 979-984. doi: 10.1139/p11-083.  Google Scholar

[26]

M. L. Wang, X. Z. Li and J. L. Zhang, The ${G'}/G$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 (2008), 417-423. doi: 10.1016/j.physleta.2007.07.051.  Google Scholar

[27]

A. M. Wazwaz, Solitons and singular solutions for the Gardner-KP equation, Applied Mathematics and Computation, 204 (2008), 162-169. doi: 10.1016/j.amc.2008.06.011.  Google Scholar

[28]

A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei and A. Biswas, New exact travelling wave solutions for DS-I and DS-II equations, Nonlinear Analysis: Modelling and Control, 17 (2012), 369-378.  Google Scholar

[29]

E. Zayed and K. A. Gepreel, Some applications of the ${G'}/G$-expansion method to non-linear partial differential equations, Applied Mathematics and Computation, 212 (2009), 1-13. doi: 10.1016/j.amc.2009.02.009.  Google Scholar

[30]

J. Zhang, F. Jiang and X. Zhao, An improved ${G'}/G$-expansion method for solving nonlinear evolution equations, International Journal of Computer Mathematics, 87 (2010), 1716-1725. doi: 10.1080/00207160802450166.  Google Scholar

show all references

References:
[1]

M. Antonova and A. Biswas, Adiabatic parameter dynamics of perturbed solitons, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 734-748. doi: 10.1016/j.cnsns.2007.12.004.  Google Scholar

[2]

A. H. Bhrawy, M. A. Abdelkawy, S. Kumar and A. Biswas, Solitons and other solutions to Kadomtsev-Petviashvili equation of B-type, Romanian Journal of Physics, 58 (2013), 729-748.  Google Scholar

[3]

A. Biswas and E. Zerrad, Soliton perturbation theory for the Gardner equation, Advanced Studies in Theoretical Physics, 2 (2008), 787-794.  Google Scholar

[4]

A. Biswas and A. Ranasinghe, 1-soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 214 (2009), 645-647. doi: 10.1016/j.amc.2009.04.001.  Google Scholar

[5]

A. Biswas and A. Ranasinghe, Topological 1-soliton solution of Kadomtsev-Petviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 217 (2010), 1771-1773. doi: 10.1016/j.amc.2009.09.042.  Google Scholar

[6]

R. Choudhury and S. K. Das, Viscelastic MHD free convective flow through porous media in presence of radiation and chemical reaction with heat and mass transfer, Journal of Applied Fluid Mechanics, 7 (2014), 603-609. Google Scholar

[7]

G. Ebadi, N. Y. Fard, A. H. Bhrawy, S. Kumar, H. Triki, A. Yildirim and A. Biswas, Solitons and other solutions to (3+1)-dimensional extended Kadomtsev-Petviashvili equation with power law nonlinearity, Romaninan Reports in Physics, 65 (2013), 27-62. Google Scholar

[8]

M. Eslami, M. Mirzazadeh and A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger's equation in optical fibers with time-dependent coefficients by simplest equation approach, Journal of Modern Optics, 60 (2013), 1627-1636. doi: 10.1080/09500340.2013.850777.  Google Scholar

[9]

M. Eslami and M. Mirzazadeh, Topological 1-soliton solution of nonlinear Schrodinger equation with dual-power law nonlinearity in nonlinear optical fibers, European Physical Journal, Plus, 128 (2013), p140. Google Scholar

[10]

E. V. Krishnan, H. Triki, M. Labidi and A. Biswas, A study of shallow water waves with Gardner's equation, Nonlinear Dynamics, 66 (2011), 497-507. doi: 10.1007/s11071-010-9928-7.  Google Scholar

[11]

S. Kundu and K. Ghoshal, An explicit model for concentration distribution using biquadratic log-wake law in an open channel flow, Journal of Applied Fluid Mechanics, 6 (2013), 339-350. Google Scholar

[12]

Z. G. Makukula and S. S. Motsa, Spectral homotopy analysis method for PDEs that model the unsteady Von Karma swirling flow, Journal of Applied Fluid Mechanics, 7 (2014), 711-718. Google Scholar

[13]

M. Mirzazadeh, M. Eslami and A. Biswas, Soliton solutions of the generalized Klein-Gordon equation by using ${G'}/G$-expansion method, Computational and Applied Mathematics, 33 (2014), 831-839. doi: 10.1007/s40314-013-0098-3.  Google Scholar

[14]

M. Mirzazadeh and M. Eslami, Exact solutions for nonlinear variants of Kadomtsev-Petviashvili (n, n) equation using functional variable method, Pramana, 81 (2013), 225-236. Google Scholar

[15]

A. Nazarzadeh, M. Eslami and M. Mirzazadeh, Exact solutions of some nonlinear partial differential equations using functional variable method, Pramana, 81 (2013), 225-236. doi: 10.1007/s12043-013-0565-9.  Google Scholar

[16]

D. Pal and S. Chatterjee, Effects of radiation on Darcey-Forchheimer convective flow over a stretching sheet in a micropolar fluid with a non-uniform heat source/sink, Journal of Applied Fluid Mechanics, 8 (2015), 207-212. Google Scholar

[17]

P. Ram and V. Kumar, Rotationally symmetric ferrofluid flow and heat transfer in porous medium with variable viscosity and viscous dissipation, Journal of Applied Fluid Mechanics, 7 (2014), 357-366. Google Scholar

[18]

S. M. Shafiof, Z. Bagheri and Sousaraei, New solutions for positive and negative Gardner-KP equations, World Applied Science Journal, 13 (2011), 662-666. Google Scholar

[19]

N. Taghizadeh and M. Mirzazadeh, The simplest equation method to study perturbed nonlinear Schrodinger's equation with Kerr law nonlinearity, Communications in Nonlinear Science and Numerical Simulations, 17 (2012), 1493-1499. doi: 10.1016/j.cnsns.2011.09.023.  Google Scholar

[20]

N. Taghizadeh, M. Mirzazadeh and F. Farahrooz, Exact soliton solutions of the modified KdV-KP equation and the Burgers-KP equation by using the first integral method, Applied Mathematical Modelling, 35 (2011), 3991-3997. doi: 10.1016/j.apm.2011.02.001.  Google Scholar

[21]

N. Taghizadeh, M. Mirzazadeh and A. Samiei Paghaleh, Exact solutions of some nonlinear evolution equations via the first integral method, Ain Shams Engineering Journal, 4 (2013), 493-499. doi: 10.1016/j.asej.2012.10.002.  Google Scholar

[22]

W. M. Taha, M. S. M. Noorani and I. Hashim, New exact solutions of sixth-order thin-film equation, Journal of King Saud University- Science, 26 (2014), 75-78. doi: 10.1016/j.jksus.2013.07.001.  Google Scholar

[23]

F. Tascan, A. Bekir and M. Koparan, Travelling wave solutions of nonlinear evolutions by using the first integral method, Communications in Nonlinear Science and Numerical Simulations, 14 (2009), 1810-1815. doi: 10.1016/j.cnsns.2008.07.009.  Google Scholar

[24]

F. Tascan and A. Bekir, Travelling wave solutions of the Cahn-Allen equation by using first integral method, Applied Mathematics and Computation, 207 (2009), 279-282. doi: 10.1016/j.amc.2008.10.031.  Google Scholar

[25]

H. Triki, B. J. M. Sturdevant, T. Hayat, O. M. Aldossary and A. Biswas, Shock wave solutions of the variants of Kadomtsev-Petviashvili equation, Canadian Journal of Physics, 89 (2011), 979-984. doi: 10.1139/p11-083.  Google Scholar

[26]

M. L. Wang, X. Z. Li and J. L. Zhang, The ${G'}/G$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A, 372 (2008), 417-423. doi: 10.1016/j.physleta.2007.07.051.  Google Scholar

[27]

A. M. Wazwaz, Solitons and singular solutions for the Gardner-KP equation, Applied Mathematics and Computation, 204 (2008), 162-169. doi: 10.1016/j.amc.2008.06.011.  Google Scholar

[28]

A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei and A. Biswas, New exact travelling wave solutions for DS-I and DS-II equations, Nonlinear Analysis: Modelling and Control, 17 (2012), 369-378.  Google Scholar

[29]

E. Zayed and K. A. Gepreel, Some applications of the ${G'}/G$-expansion method to non-linear partial differential equations, Applied Mathematics and Computation, 212 (2009), 1-13. doi: 10.1016/j.amc.2009.02.009.  Google Scholar

[30]

J. Zhang, F. Jiang and X. Zhao, An improved ${G'}/G$-expansion method for solving nonlinear evolution equations, International Journal of Computer Mathematics, 87 (2010), 1716-1725. doi: 10.1080/00207160802450166.  Google Scholar

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