Browse > Article
http://dx.doi.org/10.5831/HMJ.2017.39.1.27

SOME EXPLICIT SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS  

Kim, Hyunsoo (Department of Mathematics, Sungkyunkwan University)
Lee, Youho (Department of Internet Information, Daegu Haany University)
Publication Information
Honam Mathematical Journal / v.39, no.1, 2017 , pp. 27-40 More about this Journal
Abstract
In this paper, we construct exact traveling wave solutions of various kind of partial differential equations arising in mathematical science by the system technique. Further, the $Painlev{\acute{e}}$ test is employed to investigate the integrability of the considered equations. In particular, we describe the behaviors of the obtained solutions under certain constraints.
Keywords
$Painlev{\acute{e}}$ test; Wave transformation; Traveling wave solution; System technique;
Citations & Related Records
연도 인용수 순위
  • Reference
1 A. S. V. Ravi Kanth and K. Aruna, Differential transform method for solving linear and non-linear systems of partial differential equations, Physics Letters, A372 (2008) 6896-6898.
2 A. S. V. Ravi Kanth and K. Aruna, Differential transform method for solving the linear and nonlinear Klein-Gordon equation, Computer Physics Communiction, (2009) 708-711.
3 H. Trikia and A. -M.Wazwaz, Dark Solitons for a Generalized Korteweg-de Vries Equation with Time-Dependent Coefficients, Z. Naturforsch, 66a (2011) 199-204.   DOI
4 Q. Wang, Y. Chen and HQ. Zhang, A new Jacobi elliptic function rational ex- pansion method and its application to (1+1)-dimensional dispersive long wave equation, Chaos Solitons Fract., 23 (2005) 477-483.   DOI
5 H. A. Abdusalam and E. S. Fahmy, Traveling wave solutions for nonlinear wave equation with dissipation and nonlinear transport term through factorizations, Int. J. Comput. Meth., bf 4(4) (2007) 645-651.   DOI
6 W. Mal iet and W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta, 54 (1996) 563-568.   DOI
7 E. J. Parkes and B. R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Computer Physics Com- munications, 98 (1996) 288-300.   DOI
8 A. Biswas, Solitary wave solution for the generalized Kawahara equation, Applied Mathematics Letters, 22 (2009) 208-210.   DOI
9 M. L. Wang, X. Li and J. Zhang, The (G'/G)-expansion method and evolution equation in mathematical physics, Phys. Lett. A, 372 (2008) 417-421.   DOI
10 A. Biswas, Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett., 22 (2009) 208-210.   DOI
11 H. Kim and R. Sakthivel, Travelling wave solutions for time-delayed nonlinear evolution equations, Applied Mathematics Letters, 23 (2010) 527-532.   DOI
12 K. A. Kudryashov, Exact solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A, 147 (1990) 287-291.   DOI
13 N. A. Kudryashov, Simplest equation method to look for exact solutions of non- linear differential equations, Chaos, Solitons & Fractals, 24 (2005) 1217-1231.   DOI
14 N. K. Vitanov, Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity, Communications in Nonlinear Science and Numerical Simulation, 15 (2010) 2050-2060.   DOI
15 H. Kim, J. -H. Bae and R. Sakthivel, Exact Travelling Wave Solutions of two Important Nonlinear Partial Differential Equations, Z. Naturforsch, 69a (2014) 155-162.
16 J. H. Choi, H. Kim and R. Sakthivel, Exact solution of the wick-type stochastic fractional coupled KdV equations, J. Math. Chem., 52 (2014) 2482-2493.   DOI
17 M. M. Kabir, A. E. Y. K. Aghdam and A. Y. Koma, Modi ed Kudryashov method for nding exact solitary wave solutions of higher order nonlinear equations, Math. Methods Appl. Sci., 34(2) (2011) 213-219.   DOI
18 S. M. Ege and E. Misirli, The modified Kudryashov method for solving some fractional-order nonlinear equations, Adv. Differ. Eq., 135(1) (2014).
19 N. A. Kudryashov and A. S. Zakharchenko, Plainleve analysis and exact solu-tions of a predator-prey system with diffusion, Math. Metho. Appl. Sci., DOI: 10.1002/mma.3156 (2014).   DOI
20 A. M. Abourabia, K. M. Hassan and E. S. Selim, Painleve test and some exact solutions for (2+1)-dimensional modi ed Korteweg-de Vries-Bergers equation, Int. J. Comput. Meth., 10(3) (2013) 1250058.   DOI
21 K. R. Raslan, Exact Solitary Wave Solutions of Equal WidthWave and Related Equations Using a Direct Algebraic Method, International Journal of Nonlinear Science, 6(3) (2008) 246-254.
22 Y. Liu, F. Duan and C. Hu, Painleve property and exact solutions to a (2+1)-dimensional KdV-mKdV equation, J. of Appl. Math. and Phys., 3(6) (2015) 36083.
23 P. J. Morrison, J. D. Meiss and J. R. Cary, Scattering of Regularized-Long-Wave Solitary Waves, Physica D. Nonlinear Phenomena, (1984) 324-336.
24 A. Bekir and E. Yusufoglu, Numerical simulation of equal-width wave equation, Computers and Mathematics with Applications, 54 (2007) 1147-1153.   DOI
25 A. Maccari, The Kadomtsev.Petviashvili Equation as a Source of Integrable Model Equations, J. Math. Phys., 37 (1996) 6207-6212.   DOI
26 S. Zhang, Exp-function method for solving Maccari's system, Phys. Lett. A, 371 (2007) 65-71.   DOI
27 N. A. Kudryashov, Painleve analysis and exact solutions of the fourth-order equa-tion for description of nonlinear waves, Commun. Nonlinear Sci. Numer. Simu- lat., 28 (2015) 1-9.   DOI