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Extended simplest equation method for nonlinear differential equations. (English) Zbl 1168.34003
The authors consider the equation
$P(y,y',y'',\dots)=0, \tag{1}$ where $$y=y(z)$$ is an unknown function, $$P$$ is a polynomial in the variable $$y$$ and its derivatives and look for exact solutions $$y=y(z)$$ of the form
$y(z)=\sum_{k=0}^NA_k\left( \frac{\psi '}{\psi} \right)^k, \tag{2}$ $$A_k= \text{const}$$, $$A_N\neq 0$$, where the function $$\psi=\psi(z)$$ is the general solution of the linear ordinary differential equation
$\psi ''' +\alpha\psi '' +\beta \psi ' +\gamma \psi=0, \tag{3}$ $$\alpha, \beta, \gamma =\text{const}$$. They propose the algorithm for searching the parameters $$N,A_k,$$ $$k=1,\dots,N$$, $$\alpha,\beta,\gamma$$. This approach for the exact solution of the equation (1) the authors call the extended simplest equation method. They apply this method to the Sharma-Tasso-Olver and the Burgers-Huxley equations. New exact solutions of these equations are obtained.

##### MSC:
 34A05 Explicit solutions, first integrals of ordinary differential equations
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##### References:
 [1] Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M., Method for solving the Korteweg-de Vries equation, Phys. rev. lett., 19, 1095-1097, (1967) · Zbl 1061.35520 [2] Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H., Method for solving the sine – gordon equation, Phys. rev. lett., 30, 1262-1264, (1973) [3] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equation and inverse scattering, (1991), Cambridge University Press New York · Zbl 0762.35001 [4] Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. rev. lett., 27, 1192-1194, (1971) · Zbl 1168.35423 [5] Kudryashov, N.A., Analytical theory of nonlinear differential equations, (2004), Institute for Computer Investigations Moskow-Igevsk, (in Russian) [6] Weiss, J.; Tabor, M.; Carnevalle, G., The painleve property for partial differential equations, J. math. phys., 24, 522-526, (1983) · Zbl 0514.35083 [7] Kudryashov, N.A., Special polynomials associated with some hierarchies, Phys. lett. A, 372, 945-1956, (2008) · Zbl 1220.34111 [8] He, J.H.; Wu, X.H., Exp-function method for nonlinear wave equations, Chaos solitons fract., 30, 700-708, (2006) · Zbl 1141.35448 [9] He, J.H.; Abdou, M.A., New periodic solutions for nonlinear evolution equations using exp-function method, Chaos solitons fract., 34, 1421-1429, (2007) · Zbl 1152.35441 [10] Ebaid, A., Exact solitary wave solutions for some nonlinear evolution equations via exp-function method, Phys. lett. A, 365, 213-219, (2007) · Zbl 1203.35213 [11] El-Wakil, S.A.; Madkour, M.A.; Abdou, M.A., Application of exp-function method for nonlinear evolution equations with variable coefficients, Phys. lett. A, 369, 62-69, (2007) · Zbl 1209.81097 [12] Abdou, M.A.; Soliman, A.A.; El-Basyony, S.T., New application of exp-function method for improved Boussinesq equation, Phys. lett. A, 369, 469-475, (2007) · Zbl 1209.81091 [13] El-Wakil, S.A.; Abdou, M.A.; Hendi, A., New periodic wave solutions via exp-function method, Phys. lett. A, 372, 830-840, (2008) · Zbl 1217.37070 [14] Zhang, S., Application of exp-function method to Riccati equation and new exact solutions with three arbitrary functions of broer – kaup – kupershmidt equations, Phys. lett. A, 372, 1873-1880, (2008) · Zbl 1220.37071 [15] Kudryashov, N.A., On types of nonlinear nonintegrable equations with exact solutions, Phys. lett. A, 155, 269-275, (1991) [16] Kudryashov, N.A., Exact solutions of the generalized kuramoto – sivashinsky equation, Phys. lett. A, 147, 287-291, (1990) [17] Parkes, E.J.; Duffy, B.R., An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. phys. commun., 98, 288-300, (1996) · Zbl 0948.76595 [18] El-Wakil, S.A.; El-Labany, S.K.; Zahran, M.A.; Sabry, R., Modified extended tanh-function method and its applications to nonlinear equations, Appl. math. comput., 161, 403-412, (2005) · Zbl 1062.35082 [19] Huber, A., Solitary solutions of some nonlinear evolution equations, Appl. math. comput., 166, 464-474, (2005) · Zbl 1080.35111 [20] Yusufogˇlu, E.; Bekir, A., A travelling wave solution to the Ostrovsky equation, Appl. math. comput., 186, 256-260, (2007) · Zbl 1110.76010 [21] Liu, S.; Fu, Z.; Liu, S.; Zhao, Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. lett. A, 289, 69-74, (2001) · Zbl 0972.35062 [22] Fu, Z.; Liu, S.; Liu, S.; Zhao, Q., New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. lett. A, 290, 72-76, (2001) · Zbl 0977.35094 [23] Kudryashov, N.A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos solitons fract., 24, 1217-1231, (2005) · Zbl 1069.35018 [24] Kudryashov, N.A., Exact solitary waves of the Fisher equation, Phys. lett. A, 342, 99-106, (2005) · Zbl 1222.35054 [25] Bekir, A.; Boz, A., Exact solutions for nonlinear evolution equations using exp-function method, Phys. lett. A, 372, 1619-1625, (2008) · Zbl 1217.35151 [26] Lan, H.; Wang, K., Exact solutions for some nonlinear equations, Phys. lett. A, 137, 369-372, (1989) [27] Ince, E.L., Ordinary differential equations, (1926), Longmans, Green & Co. London · Zbl 0063.02971 [28] Kamke, E., Differentialgleichungen, Lösungsmethoden und Lösungen I: gewöhnliche differentialgleichungen, (1943), Geest & Portig Leipzig · Zbl 0028.22702 [29] Kudryashov, N.A.; Demina, M.V., Polygons of differential equations for finding exact solutions, Chaos solitons fract., 33, 1480-1496, (2007) · Zbl 1133.35084
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