# American Institute of Mathematical Sciences

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

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

 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.
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.  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.   Google Scholar [3] A. Biswas and E. Zerrad, Soliton perturbation theory for the Gardner equation,, Advanced Studies in Theoretical Physics, 2 (2008), 787.   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.  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.  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.   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.   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.  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, 128 (2013).   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.  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.   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.   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.  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.   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.  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.   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.   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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  doi: 10.1080/00207160802450166.  Google Scholar

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##### References:
 [1] M. Antonova and A. Biswas, Adiabatic parameter dynamics of perturbed solitons,, Communications in Nonlinear Science and Numerical Simulation, 14 (2009), 734.  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.   Google Scholar [3] A. Biswas and E. Zerrad, Soliton perturbation theory for the Gardner equation,, Advanced Studies in Theoretical Physics, 2 (2008), 787.   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.  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.  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.   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.   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.  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, 128 (2013).   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.  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.   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.   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.  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.   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.  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.   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.   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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  doi: 10.1080/00207160802450166.  Google Scholar
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