Painlevé analysis and exact solutions of the Korteweg-de Vries equation with a source

Abstract We consider the Korteweg–de Vries equation with a source. The source depends on the solution as polynomials with constant coefficients. Using the Painleve test we show that the generalized Korteweg–de Vries equation is not integrable by the inverse scattering transform. However there are some exact solutions of the generalized Korteweg–de Vries equation for two forms of the source. We present these exact solutions.

[1]  Nikolai A. Kudryashov,et al.  A note on the G'/G-expansion method , 2010, Appl. Math. Comput..

[2]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[3]  Yuri S. Kivshar,et al.  Dynamics of Solitons in Nearly Integrable Systems , 1989 .

[4]  W. Hereman,et al.  The tanh method: I. Exact solutions of nonlinear evolution and wave equations , 1996 .

[5]  N. A. Kudryashov.  Simplest equation method to look for exact solutions of nonlinear differential equations , 2005 .

[6]  Nikolai A. Kudryashov,et al.  A note on solutions of the generalized Fisher equation , 2014, Appl. Math. Lett..

[7]  Nikolai A. Kudryashov,et al.  One method for finding exact solutions of nonlinear differential equations , 2011, 1108.3288.

[8]  Nikolai A. Kudryashov,et al.  On "new travelling wave solutions" of the KdV and the KdV-Burgers equations , 2009 .

[9]  Nikolai A. Kudryashov,et al.  Polynomials in logistic function and solitary waves of nonlinear differential equations , 2013, Appl. Math. Comput..

[10]  A. Polyanin,et al.  Handbook of Nonlinear Partial Differential Equations , 2003 .

[11]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[12]  Anjan Biswas,et al.  Solitary wave solution for the generalized Kawahara equation , 2009, Appl. Math. Lett..

[13]  D. Korteweg,et al.  On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 2011 .

[14]  Norman J. Zabusky,et al.  Soliton , 2010, Scholarpedia.

[15]  MalflietW..  The tanh method , 2004 .

[16]  Nikolay K. Vitanov,et al.  Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity , 2010 .

[17]  Anjan Biswas,et al.  Modified simple equation method for nonlinear evolution equations , 2010, Appl. Math. Comput..