Painlevé analysis and exact solutions of the fourth-order equation for description of nonlinear waves

• Nonlinear evolution equations of the fourth order are studied.
• Painlevé analysis to equations is used.
• Exact solutions of the fourth order equation are found.

The fourth-order equation for description of nonlinear waves is considered. A few variants of this equation are studied. Painlevé test is applied to investigate integrability of these equations. We show that all these equations are not integrable, but some exact solutions of these equations exist. Analytic solutions in closed-form of the equations are found.