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Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a source. (English) Zbl 1321.35203
The paper deals with the Korteweg-de Vries equation with a source in the form of a polynomial with respect to the solution, with constant coefficients. The integrability of such equations is studied using the Painlevé test applied to the traveling wave solutions. It is shown that the first step of the test is not passed for polynomials of degree \(n=3\) and \(n\geq 5\). As for \(n=2\) and \(n=4\), some explicit solutions corresponding to these cases are given by using the method of \(Q\)-functions (satisfying the Riccati equation).

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35C05 Solutions to PDEs in closed form
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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[1] N.A. Kudryashov, Methods of nonlinear mathematical physics, Dolgoprudny, Intellect, 468 p. (in Russian).
[2] Ablowitz, M. J.; Clarkson, P. A., (Soliton, Nonlinear Evolution Equations and Iverse Scattering, London Mathematical Society Lecture Note Series, vol. 149, (1991), Cambridge University Press), 516 p · Zbl 0762.35001
[3] Korteweg, D. J.; de Vies, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long ststionary waves, Phil. Mag., 39, 5, 43-422, (1895) · JFM 26.0881.02
[4] Kudryashov, N. A., On “new travelling wave solutions” of the KdV and the KdV-Burgers equtions, Commun. Nonlinear Sci. Numer. Simul., 14, 5, 1891-1900, (2009) · Zbl 1221.35343
[5] Kivshar, Yu. S.; Malomed, B. A., Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys., 61, 4, 763-915, (1989)
[6] Polyanin, A. D.; Zaitsev, V. F., Handbook of nonlinear partial differential equations, (2011), Chapman and Hall/CRC Boca Raton · Zbl 1243.35001
[7] Malfliet, W.; Hereman, W., The tanh method: I. exact solutions of nonlinear evolution and wave equations, Phys. Scripta., 54, 563-568, (1996) · Zbl 0942.35034
[8] Biswas, A., Solitary wave solution for the generalized Kawahara equation, Appl. Math. Lett., 22, 208-210, (2009) · Zbl 1163.35468
[9] Kudryashov, N. A., Simplest equation method to look for exact solutions of nonlinear differential equations, Chaos Solitons Fractals, 24, 1217-1231, (2005) · Zbl 1069.35018
[10] Vitanov, N. K., Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity, Commun. Nonlinear Sci. Numer. Simul., 15, 2050-2060, (2010) · Zbl 1222.35062
[11] Jaward, A. J.M.; Petkovic, M. D.; Biswas, A., Modified simple equation method for nonlinear evolution equations, Appl. Math. Comput., 217, 869-877, (2010) · Zbl 1201.65119
[12] Wang, M. L.; Li, X.; Zhang, J., The G’/G-expansion method and evolution equation in mathematical physics, Phys. Lett. A., 372, 417-421, (2008)
[13] Kudryashov, N. A., A note on the G’/G-expansion method, Appl. Math. Comput., 217, 1755-1758, (2010) · Zbl 1203.35228
[14] Kudryashov, N. A., One method for finding exact solutions of nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul., 17, 2248-2253, (2012) · Zbl 1250.35055
[15] Kudryashov, N. A., Polynomials in logistic function and solitary waves of nonlinear differential equations, Appl. Math. Comput., 219, 9245-9253, (2013) · Zbl 1297.35076
[16] Kudryashov, N. A.; Zakharchenko, A. S., A note on solutions of the generalized Fisher equation, Appl. Math. Lett., 32, 53-56, (2014) · Zbl 1327.35165
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