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Dynamics of shallow water waves with GardnerKadomtsevPetviashvili equation
1.  Computer Engineering Technique Department AlRafidain, University College, Baghdad, Iraq 
2.  Department of Engineering Sciences, Faculty of Technology and Engineering East of Guilan, University of Guilan, P.C. 4489163157, RudsarVajargah, Iran 
3.  Department of Mathematical Sciences, Delaware State University, Dover, DE 199012277, United States 
References:
[1] 
M. Antonova and A. Biswas, Adiabatic parameter dynamics of perturbed solitons, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 734748. doi: 10.1016/j.cnsns.2007.12.004. 
[2] 
A. H. Bhrawy, M. A. Abdelkawy, S. Kumar and A. Biswas, Solitons and other solutions to KadomtsevPetviashvili equation of Btype, Romanian Journal of Physics, 58 (2013), 729748. 
[3] 
A. Biswas and E. Zerrad, Soliton perturbation theory for the Gardner equation, Advanced Studies in Theoretical Physics, 2 (2008), 787794. 
[4] 
A. Biswas and A. Ranasinghe, 1soliton solution of KadomtsevPetviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 214 (2009), 645647. doi: 10.1016/j.amc.2009.04.001. 
[5] 
A. Biswas and A. Ranasinghe, Topological 1soliton solution of KadomtsevPetviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 217 (2010), 17711773. doi: 10.1016/j.amc.2009.09.042. 
[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), 603609. 
[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 KadomtsevPetviashvili equation with power law nonlinearity, Romaninan Reports in Physics, 65 (2013), 2762. 
[8] 
M. Eslami, M. Mirzazadeh and A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger's equation in optical fibers with timedependent coefficients by simplest equation approach, Journal of Modern Optics, 60 (2013), 16271636. doi: 10.1080/09500340.2013.850777. 
[9] 
M. Eslami and M. Mirzazadeh, Topological 1soliton solution of nonlinear Schrodinger equation with dualpower law nonlinearity in nonlinear optical fibers, European Physical Journal, Plus, 128 (2013), p140. 
[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), 497507. doi: 10.1007/s1107101099287. 
[11] 
S. Kundu and K. Ghoshal, An explicit model for concentration distribution using biquadratic logwake law in an open channel flow, Journal of Applied Fluid Mechanics, 6 (2013), 339350. 
[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), 711718. 
[13] 
M. Mirzazadeh, M. Eslami and A. Biswas, Soliton solutions of the generalized KleinGordon equation by using ${G'}/G$expansion method, Computational and Applied Mathematics, 33 (2014), 831839. doi: 10.1007/s4031401300983. 
[14] 
M. Mirzazadeh and M. Eslami, Exact solutions for nonlinear variants of KadomtsevPetviashvili (n, n) equation using functional variable method, Pramana, 81 (2013), 225236. 
[15] 
A. Nazarzadeh, M. Eslami and M. Mirzazadeh, Exact solutions of some nonlinear partial differential equations using functional variable method, Pramana, 81 (2013), 225236. doi: 10.1007/s1204301305659. 
[16] 
D. Pal and S. Chatterjee, Effects of radiation on DarceyForchheimer convective flow over a stretching sheet in a micropolar fluid with a nonuniform heat source/sink, Journal of Applied Fluid Mechanics, 8 (2015), 207212. 
[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), 357366. 
[18] 
S. M. Shafiof, Z. Bagheri and Sousaraei, New solutions for positive and negative GardnerKP equations, World Applied Science Journal, 13 (2011), 662666. 
[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), 14931499. doi: 10.1016/j.cnsns.2011.09.023. 
[20] 
N. Taghizadeh, M. Mirzazadeh and F. Farahrooz, Exact soliton solutions of the modified KdVKP equation and the BurgersKP equation by using the first integral method, Applied Mathematical Modelling, 35 (2011), 39913997. doi: 10.1016/j.apm.2011.02.001. 
[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), 493499. doi: 10.1016/j.asej.2012.10.002. 
[22] 
W. M. Taha, M. S. M. Noorani and I. Hashim, New exact solutions of sixthorder thinfilm equation, Journal of King Saud University Science, 26 (2014), 7578. doi: 10.1016/j.jksus.2013.07.001. 
[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), 18101815. doi: 10.1016/j.cnsns.2008.07.009. 
[24] 
F. Tascan and A. Bekir, Travelling wave solutions of the CahnAllen equation by using first integral method, Applied Mathematics and Computation, 207 (2009), 279282. doi: 10.1016/j.amc.2008.10.031. 
[25] 
H. Triki, B. J. M. Sturdevant, T. Hayat, O. M. Aldossary and A. Biswas, Shock wave solutions of the variants of KadomtsevPetviashvili equation, Canadian Journal of Physics, 89 (2011), 979984. doi: 10.1139/p11083. 
[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), 417423. doi: 10.1016/j.physleta.2007.07.051. 
[27] 
A. M. Wazwaz, Solitons and singular solutions for the GardnerKP equation, Applied Mathematics and Computation, 204 (2008), 162169. doi: 10.1016/j.amc.2008.06.011. 
[28] 
A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei and A. Biswas, New exact travelling wave solutions for DSI and DSII equations, Nonlinear Analysis: Modelling and Control, 17 (2012), 369378. 
[29] 
E. Zayed and K. A. Gepreel, Some applications of the ${G'}/G$expansion method to nonlinear partial differential equations, Applied Mathematics and Computation, 212 (2009), 113. doi: 10.1016/j.amc.2009.02.009. 
[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), 17161725. doi: 10.1080/00207160802450166. 
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), 734748. doi: 10.1016/j.cnsns.2007.12.004. 
[2] 
A. H. Bhrawy, M. A. Abdelkawy, S. Kumar and A. Biswas, Solitons and other solutions to KadomtsevPetviashvili equation of Btype, Romanian Journal of Physics, 58 (2013), 729748. 
[3] 
A. Biswas and E. Zerrad, Soliton perturbation theory for the Gardner equation, Advanced Studies in Theoretical Physics, 2 (2008), 787794. 
[4] 
A. Biswas and A. Ranasinghe, 1soliton solution of KadomtsevPetviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 214 (2009), 645647. doi: 10.1016/j.amc.2009.04.001. 
[5] 
A. Biswas and A. Ranasinghe, Topological 1soliton solution of KadomtsevPetviashvili equation with power law nonlinearity, Applied Mathematics and Computation, 217 (2010), 17711773. doi: 10.1016/j.amc.2009.09.042. 
[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), 603609. 
[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 KadomtsevPetviashvili equation with power law nonlinearity, Romaninan Reports in Physics, 65 (2013), 2762. 
[8] 
M. Eslami, M. Mirzazadeh and A. Biswas, Soliton solutions of the resonant nonlinear Schrodinger's equation in optical fibers with timedependent coefficients by simplest equation approach, Journal of Modern Optics, 60 (2013), 16271636. doi: 10.1080/09500340.2013.850777. 
[9] 
M. Eslami and M. Mirzazadeh, Topological 1soliton solution of nonlinear Schrodinger equation with dualpower law nonlinearity in nonlinear optical fibers, European Physical Journal, Plus, 128 (2013), p140. 
[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), 497507. doi: 10.1007/s1107101099287. 
[11] 
S. Kundu and K. Ghoshal, An explicit model for concentration distribution using biquadratic logwake law in an open channel flow, Journal of Applied Fluid Mechanics, 6 (2013), 339350. 
[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), 711718. 
[13] 
M. Mirzazadeh, M. Eslami and A. Biswas, Soliton solutions of the generalized KleinGordon equation by using ${G'}/G$expansion method, Computational and Applied Mathematics, 33 (2014), 831839. doi: 10.1007/s4031401300983. 
[14] 
M. Mirzazadeh and M. Eslami, Exact solutions for nonlinear variants of KadomtsevPetviashvili (n, n) equation using functional variable method, Pramana, 81 (2013), 225236. 
[15] 
A. Nazarzadeh, M. Eslami and M. Mirzazadeh, Exact solutions of some nonlinear partial differential equations using functional variable method, Pramana, 81 (2013), 225236. doi: 10.1007/s1204301305659. 
[16] 
D. Pal and S. Chatterjee, Effects of radiation on DarceyForchheimer convective flow over a stretching sheet in a micropolar fluid with a nonuniform heat source/sink, Journal of Applied Fluid Mechanics, 8 (2015), 207212. 
[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), 357366. 
[18] 
S. M. Shafiof, Z. Bagheri and Sousaraei, New solutions for positive and negative GardnerKP equations, World Applied Science Journal, 13 (2011), 662666. 
[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), 14931499. doi: 10.1016/j.cnsns.2011.09.023. 
[20] 
N. Taghizadeh, M. Mirzazadeh and F. Farahrooz, Exact soliton solutions of the modified KdVKP equation and the BurgersKP equation by using the first integral method, Applied Mathematical Modelling, 35 (2011), 39913997. doi: 10.1016/j.apm.2011.02.001. 
[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), 493499. doi: 10.1016/j.asej.2012.10.002. 
[22] 
W. M. Taha, M. S. M. Noorani and I. Hashim, New exact solutions of sixthorder thinfilm equation, Journal of King Saud University Science, 26 (2014), 7578. doi: 10.1016/j.jksus.2013.07.001. 
[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), 18101815. doi: 10.1016/j.cnsns.2008.07.009. 
[24] 
F. Tascan and A. Bekir, Travelling wave solutions of the CahnAllen equation by using first integral method, Applied Mathematics and Computation, 207 (2009), 279282. doi: 10.1016/j.amc.2008.10.031. 
[25] 
H. Triki, B. J. M. Sturdevant, T. Hayat, O. M. Aldossary and A. Biswas, Shock wave solutions of the variants of KadomtsevPetviashvili equation, Canadian Journal of Physics, 89 (2011), 979984. doi: 10.1139/p11083. 
[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), 417423. doi: 10.1016/j.physleta.2007.07.051. 
[27] 
A. M. Wazwaz, Solitons and singular solutions for the GardnerKP equation, Applied Mathematics and Computation, 204 (2008), 162169. doi: 10.1016/j.amc.2008.06.011. 
[28] 
A. Yildirim, A. Samiei Paghaleh, M. Mirzazadeh, H. Moosaei and A. Biswas, New exact travelling wave solutions for DSI and DSII equations, Nonlinear Analysis: Modelling and Control, 17 (2012), 369378. 
[29] 
E. Zayed and K. A. Gepreel, Some applications of the ${G'}/G$expansion method to nonlinear partial differential equations, Applied Mathematics and Computation, 212 (2009), 113. doi: 10.1016/j.amc.2009.02.009. 
[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), 17161725. doi: 10.1080/00207160802450166. 
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