Common errors in finding exact solutions of nonlinear equations (original) (raw)

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These errors often arise from straightforward application of common computer algebra systems, such as Maple or Mathematical, which can generate large lists of exact solutions. Some authors believe that they obtain new exact solutions by using such systems. However, many of these "new" solutions are equivalent to one another and can be reduced to a single well-known solution, which is revealed by detailed investigation. Furthermore, it is not uncommon to come across "new" solutions that simply do not solve the original equation—authors perform a number of cumbersome intermediate calculations and forget to verify the result. These and other common errors are addressed in the papers listed below. N. A. Kudryashov, Seven common errors in finding exact solutions of nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 9-10, pp. 3507-3529, 2009. (html, pdf) N. A. Kudryashov, On "new travelling wave solutions" of the KdV and the KdV-Burgers equations, Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 5, pp. 1891-1900, 2009. (pdf) N. A. Kudryashov and N. B. Loguinova, Be careful with the Exp-function method, Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 5, pp. 1881-1890, 2009. (pdf) N. A. Kudryashov, Comment on: "A novel approach for solving the Fisher equation using Exp-function method" [Phys. Lett. A, 372 (2008), 3836-3840], Physics Letters A, Vol. 373, No. 12-13, pp. 1196-1197, 2009. (pdf) N. A. Kudryashov and M. B. Soukharev, Popular ansatz methods and solitary waves solutions of the Kuramoto-Sivashinsky equation, Regular and Chaotic Dynamics, Vol. 14, No. 3, pp. 407-419, 2009. (pdf) N. A. Kudryashov, A note on new exact solutions for the Kawahara equation using Exp-function method, Journal of Computational and Applied Mathematics, Vol. 234, No. 12, pp. 3511-3512, 2010. (pdf) N. A. Kudryashov, Unnecessary exact solutions of nonlinear ordinary differential equations, arXiv:1011.1623v1 [nlin,SI] 7 Nov 2010. (pdf) N. A. Kudryashov, P. N. Ryabov, D. I. SinelshchikovA note on "New kink-shaped solutions and periodic wave solutions for the (2+1)-dimensional Sine-Gordon equation", Applied Mathematics and Computation, Vol. 216, No. 8, pp. 2479-2481, 2010. (pdf). N. A. Kudryashov, M. B. SoukharevComment on: "Multi soliton solution, rational solution of the Boussinesq-Burgers equations", Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 7, pp. 1765-1767, 2010. (pdf). N. A. KudryashovMeromorphic solutions of nonlinear ordinary differential equations, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 10, pp. 2778-2790, 2010. (pdf). N. A. KudryashovA Note on New exact solutions for the Kawahara equation using Exp-function method, Journal of Computational and Applied Mathematics, Vol. 234, No. 12, pp. 3511-3512, 2010. (pdf). N. A. KudryashovA note on the G'/G - expansion method, Applied Mathematics and Computation, Vol. 217, No. 4, pp. 1755-1758, 2010. (pdf). N. A. KudryashovA new note on exact complex travelling wave solutions for (2+1) - dimensional B-type Kadomtsev-Petviashvili equation, Applied Mathematics and Computation, Vol. 217, No. 5, pp. 2282-2284, 2010. (pdf). N. A. Kudryashov, D. I. SinelshchikovA note on "Abundant new exact solutions for the (3+1)-dimensional Jimbo-Miwa equation", Journal of Mathematical Analysis and Applications, Vol. 371, No. 1, pp. 393-396, 2010. (pdf). N. A. Kudryashov, D. I. SinelshchikovA Note on the Lie Symmetry Analysis and Exact Solutions for the Extended mKdV Equation, Acta Applicandae Mathematicae, Vol. 113, No. 1, pp. 41-44, 2011. (pdf). N. A. Kudryashov, P. N. Ryabov, D. I. SinelshchikovComment on: "Application of the (G'/G) method for the complex KdV equation" [Huiqun Zhang, Commun Nonlinear Sci Numer Simulat 15, 2010, 1700-1704], Communications in Nonlinear Science and Numerical Simulations, Vol. 16, No. 1, pp. 596-598, 2011. (pdf) N. A. Kudryashov A Note on solutions of the Korteweg-de Vries hierarchy, Communications in Nonlinear Science and Numerical Simulations, Vol. 16, No. 4, pp. 1703-1705, 2011. (pdf) N. A. Kudryashov On Completely Integrability Systems of Differential Equations, Communications in Nonlinear Science and Numerical Simulations, Vol. 16, No. 6, pp. 2414-2420, 2011. (pdf) N. A. Kudryashov Redundant exact solutions of nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulations, 2011. 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