On solutions of generalized modified Korteweg–de Vries equation of the fifth order with dissipation

Abstract

The generalized modified Korteweg–de Vries equation of the fifth order with dissipation is considered. The Painlevé test is applied for studying integrability of this equation. It is shown that the generalized modified Korteweg–de Vries equation of the fifth order does not pass the Painlevé test in the general case but has the expansion of the solution in the Laurent series. As a consequence the equation can have some exact solutions at additional conditions on the parameters of the equation. We present the effective modification of methods for finding of solitary wave and elliptic solutions of nonlinear differential equations. Solitary wave and elliptic solutions of the generalized modified Korteweg–de Vries equation of the fifth order are found by means of expansion for solution in the Laurent series. These solutions can be used for description of nonlinear waves in the medium with dissipation, dispersion.