New exact solutions for generalized Gardner equation
Keywords:
Generalized Gardner equation, extended trial equation method, soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions.Abstract
In this research, we seek exact solutions of generalized Gardner equation. We make use of extended trial equationmethod to attain exact solutions of generalized Gardner equation. Firstly, we find some exact solutions such as soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of generalized Gardner equation by the help of extended trial equation method. Then, for some parameters, we draw two and three dimensional graphics of imaginary and real values of some exact solutions that we acquired by using this method.
References
Abdou, M.A. (2010). New periodic solitary wave solutions for a
variable-coefficient gardner equation from fluid dynamics and plasma physics. Applied Mathematics, 1:307-311.
Akter, S., Roshid, H.O., Alam, Md. N., Rahman, N. & Akbar,
M.A. (2014). Application of exp -expansion method to find
the exact solutions of nonlinear evolution equations. IOSR Journal of Mathematics, 9(6):106-113.
Alejo, M.A. (2012). Well-posedness and stability results for the
Gardner equation. Nonlinear Differential Equations and Applications, 19:503-520.
Bulut, H., Pandir, Y. & Tuluce Demiray, S. (2014a). Exact solutions
of time-fractional KdV Equations by using generalized Kudryashov
method. International Journal of Modeling and Optimization, 4(4):315-320.
Bulut, H. (2013). Classification of exact solutions for generalized
form of K(m,n) equation. Abstract and Applied Analysis, 2013, Article ID:742643, http://dx.doi.org/ 10.1155/2013/742643, 11 pages.
Bulut, H., Pandir, Y. & Tuluce Demiray, S. (2014b). Exact solutions
of nonlinear Schrodingers equation with dual power-law nonlinearity by extended trial equation method. Waves in Random and Complex Media, 24(4):439-451.
Daoui, A.K. & Triki, H. (2014). Solitary waves, shock waves and
singular solitons of Gardners equation for shallow water dynamics.
Acta Physica Polonica, B 45(6):1135-1145.
Hamdi, S., Morse, B., Halphen, B. & Schiesser, W. (2011). Analytical solutions of long nonlinear internal waves: Part I. Natural Hazards, 57:597-607.
Hong, B. & Lu, D. (2012). New exact solutions for the generalized
variable-coefficient Gardner equation with forcing term. Applied
Mathematics and Computation, 219:2732-2738.
Islam, Md. E., Khan, K., Akbar, M.A. & Islam, R. (2013). Traveling
wave solutions of nonlinear evolution equation via enhanced
-expansion method. GANIT: Journal of Bangladesh Mathematical
Society, 33:83-92.
Jawad, A.J.M. (2012). New exact solutions of nonlinear partial
differential equations using Tan-Cot function method. Studies in
Mathematical Sciences, 5(2):13-25.
Khan, K. & Akbar, M.A. (2013a). Application of Exp
-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation. World Applied Sciences Journal, 24(10):1373-1377.
Khan, K. & Akbar, M.A. (2013b). Exact and solitary wave
solutions for the Tzitzeica-Dodd-Bullough and the modified KdVZakharov-Kuznetsov equations using the modified simple equation method. Ain Shams Engineering Journal, http://dx.doi.org/10.1016/j.asej.2013.01.010, 4(4):903-909.
Khan, K., Akbar, M.A. & Alam, Md. N. (2013). Traveling wave
solutions of the nonlinear Drinfeld-Sokolov-Wilson equation and
modified Benjamin-Bona-Mahony equations. Journal of the Egyptian
Mathematical Society, http://dx.doi.org/10.1016/j.joems.2013.04.010, 21:233-240.
Khan, K. & Akbar, M.A. (2014a). Traveling wave solutions of the
(2+1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method. Ain Shams Engineering Journal, http://dx.doi.org/ 10.1016/j.asej.2013.07.007, 5(1):247-256.
Khan, K. & Akbar, M.A. (2014b). Exact solutions of the (2+1)-
dimensional cubic Klein-Gordon equation and the (3+1)-dimensional
Zakharov-Kuznetsov equation using the modified simple equation
method. Journal of the Association of Arab Universities for Basic and Applied Sciences, http://dx.doi.org/10.1016/j.jaubas.2013.05.001, 15:74-81.
Khan, K., Akbar, M.A. & Roshid, H.O. (2014). Exact traveling
wave solutions of nonlinear evolution equation via enhanced
-expansion method. British Journal of Mathematics & Computer
Science, 4(10):1318-1334.
Kumar, H., Malik, A. & Chand, F. (2013). Soliton solutions of
some nonlinear evolution equations with time-dependent coefficients.
Pramana- Journal of Physics, 80(2):361-367.
Li, W. & Ma, X.J. (2011). Exact solutions for a generalized variable
coefficient gardner equation. Applied Mathematical Sciences,
(73):3607-3618.
Li, X. & Wang, M. (2007). A sub-ODE method for finding exact
solutions of a generalized KdV-mKdV equation with high-order
nonlinear terms. Physics Letters A, 361:115-118.
Lu, D. & Liu, C. (2010). A sub-ODE method for generalized Gardner
and BBM equation with nonlinear terms of any order. Applied
Mathematics and Computation, 217:1404-1407.
Miura, R.M. (1968). Korteweg-de vries equation and generalizations I. A remarkable explicit nonlinear transformation. Journal of Mathematical Physics, 9:1202-1204.
Miura, R.M., Gardner, C.S. & Kruskal, M.D. (1968). Korteweg-de
vries equation and generalizations II. Existence of conservation laws
and constants of motion. Journal of Mathematical Physics, 9:1204-
Pandir, Y., Gurefe, Y., Kadak, U. & Misirli, E. (2012). Classification
of exact solutions for some nonlinear partial differential equations with generalized evolution. Abstract and Applied Analysis, 2012:1-12.
Pandir, Y., Gurefe, Y. & Misirli, E. (2013). The extended trial
equation method for some time fractional differential equations.
Discrete Dynamics in Nature and Society, Article ID:491359, http://
dx.doi.org/10.1155/2013/491359, 13 pages.
Rageh, T.M., Salem, G. & El-Salam, F.A. (2014). Restrictive taylor
approximation for Gardner and KdV equations. International Journal of Advances in Applied Mathematics and Mechanics, 1(3):1-10.
Tang, Y., Xu, W., Shen, J. & Gao, L. (2008). Persistence of solitary
wave solutions of singularly perturbed Gardner equation. Chaos,
Solitons and Fractals, http://dx.doi.org/10.1016/j.chaos.2006.09.044,
(2):532-538.
Tuluce Demiray, S., Pandir, Y. & Bulut, H. (2014a). Generalized
Kudryashov method for time-fractional differential equations. Abstract and Applied Analysis, 2014 Article ID: 901540 13 pages.
Tuluce Demiray, S., Pandir, Y. & Bulut, H. (2014b). The investigation of exact solutions of nonlinear time-fractional Klein-Gordon equation by using generalized Kudryashov method. AIP Conference Proceedings, 1637:283-289. Tuluce Demiray, S. & Bulut, H. (2015). Some exact solutions of
generalized Zakharov system. Waves in Random and Complex Media,25(1):75-90.
Usman, M., Zubair, T., Rashid, I. & Mohyud-Din, S.T. (2013). New
solitary wave solutions of modified KdV and Gardner equations by
U-expansion method. International Journal of Modern Mathematical
Sciences, 6(1):33-44.
Vaneeva, O., Kuriksha, O. & Sophocleous, C. (2015). Enhanced group classification of Gardner equations with time-dependent coefficients. Communications in Nonlinear Science and Numerical Simulation,
:1243-1251.
Vassilev, V.M., Djondjorov, P.A., Hadzhilazova, M.T. & Mladenov,
I.M. (2011). Traveling wave solutions of the Gardner equation and
motion of plane curves governed by the mKdV flow. AIP Conference
Proceedings, 1404:86-93.
Wazwaz, A.M. (2010). A study on KdV and Gardner equations with
time-dependent coefficients and forcing terms. Applied Mathematics and Computation, 217:2277-2281.
Wazwaz, A.M. (2007). New solitons and kink solutions for the Gardner equation. Communications in Nonlinear Science and Numerical Simulation, 12(8):1395-1404.
Yang, L.Y. (2012). Auxiliary equation method for generalized Gardner equation and BBM equation with nonlinear terms of any order. Journal of Inner Mongolia Normal University, 41(5):456-461.
Zayed, E. & Abdelaziz, M. (2011). Exact traveling wave solutions of
nonlinear variable coefficients evolution equations with forced terms
using the generalized G/G-expansion method. Wseas Transactions on Mathematics, 10(3):115-124.
Zayed, E.M. & Abdelaziz, M.A.M. (2010). Exact traveling wave
solutions of variable coefficients nonlinear Pdes using the Tanhfunction method with a generalized wave transformation. Global Journal of Science Frontier Research, http://journalofscience.org/index.
php/GJSFR/article/view/181, 10(6):8-13.
Zhang, S., Wang, W. & Tong, J.L. (2008). The improved sub-ODE
method for a generalized KdV-mKdV equation with nonlinear terms of any order. Physics Letters A, 372:3808-3813.
Zheng, B. (2011). A new Bernoulli sub-ODE method for constructing travelling wave solutions for two nonlinear equations with any order. University Politechnica of Bucharest Scientific Bulletin Series A,
(3):85-94.