The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method
Applied and Computational Mathematics
Volume 4, Issue 5, October 2015, Pages: 335-341
Received: Jun. 29, 2015;
Accepted: Aug. 5, 2015;
Published: Aug. 19, 2015
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Sen-Yung Lee, Department of Mechanical Engineering, National Cheng Kung University, Taiwan, R.O.C.
Chun-Ku Kuo, Department of Mechanical Engineering, Air Force Institute of Technology, Taiwan, R.O.C.
The general form of linearized exact solution for the Korteweg and de Vries (KdV) equation, with an arbitrary nonlinear coefficient, is derived by the simplest equation method with the Bernoulli equation as the simplest equation. It is shown that the proposed exact solution overcomes the long existing problem of discontinuity and can be successfully reduced to linearity, while the nonlinear term coefficient approaches zero. Comparison of four different soliton solutions is presented. A new phenomenon, named soliton sliding, is observed.
The General Form of Linearized Exact Solution for the KdV Equation by the Simplest Equation Method, Applied and Computational Mathematics.
Vol. 4, No. 5,
2015, pp. 335-341.
Bullough, RK, Caudrey, PJ. Solitons. New York: Springer-Verlag Berlin Heidelberg (1980)
Korteweg, DJ, G. de Vries. On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary waves. Phil. Mag. 39.5, 422-443(1895)
Wazwaz, AM. Partial differential equations and solitary waves theory. The USA: Springer (2007)
Jeffrey, A, Kakutani, T. Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation. SIAM Review. 14.4, 582-636 (1972)
Soliman, AA. The modified extended tanh-function method for solving Burgers-type equations. Physica A 361, 394-404 (2006)
Wazwaz, AM. The tanh method for travelling wave solutions of nonlinear equations. Applied Mathematics and Computation 154, 713-723 (2004)
Wazwaz, AM. The tanh method and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants. Communications in Nonlinear Science and Numerical Simulation 11, 148-160 (2006)
Hirota, R. The direct method in soliton theory. Cambridge, Cambridge University Press (2004)
Hereman, W, Nuseir, A. Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math. Comput. Simulation. 43, 13-27 (1997)
Ebaid, A. Exact solitary wave for some nonlinear evolution equations via Exp-function method. Phys. Lett. A 364, 213-219 (2007)
Liu, S, et al. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letter A 289, 69-74 (2001)
Shen, S, Pan, Z. A note on the Jacobi elliptic function expansion method. Physics Letter A 309, 143-148 (2003)
Wang, M, Wang, Y, Yubin, Z. An auto-Backlund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications. Physics Letter A 303, 45-51 (2002)
Kudryashov, NA. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simulat 17, 2248-2253 (2012)
Vitanov, NK. On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs:The role of the simplest equation. Commun Nonlinear Sci Numer Simulat 16, 4215-4231 (2011)
Kudryashov, NA. Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 14, 3507-3529 (2009)
Vitanov, NK. Application of simplest equations of Bernoulli and Riccati kind for obtaining exact travelling-wave solutions for a class of PDEs with polynominal nonlinearity. Commun Nonlinear Sci Numer Simulat 15, 2050-2060 (2010)
Kudryashov, NA. Modified method of simplest equation:Powerful tool for obtaining exact and approximate travelling-wave solutions of nonlinear PDEs. Commun Nonlinear Sci Numer Simulat 16, 1176-1185 (2011)
Kudryashov, NA, Loguinova, NB. Extended simplest equation method for nonlinear differential equations. Applied Mathematics and Computation 205, 396-402 (2008)
Mohamad, JA, Petkovic, MD, Biswas, A. Modified simple equation method for nonlinear evolution equations. Applied Mathematics and Computation 217, 869-877 (2010)
Lu, D, Hong, B, Tian L. New solitary wave and periodic wave solutions for general types of KdV and KdV-Burgers equations. Commun Nonlinear Sci Numer Simulat 14, 77-84 (2009)
Wazzan, L. A modified tanh-coth method for solving the KdV and KdV-Burgers’ equations. Commun Nonlinear Sci Numer Simulat 14, 443-450 (2009)
Spiegel, MR. Mathematical handbook of formulas and tables. McGRAW-Hill; (1968)
Wazwaz, AM. Two reliable methods for solving variants of the KdV equation with compact and noncompact structures. Chaos, Solitons and Fractals 28, 454-462 (2006)