Painlevé Test, First Integrals and Exact Solutions of Nonlinear Dissipative Differential Equations

The Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid are considered. The Cauchy problems for all these partial differential equations are not solved by the inverse scattering transform. Reductions of these equations to nonlinear ordinary differential equations do not pass the Painlevé test. However, there are local expansions of the general solutions in the Laurent series near movable singular points. We demonstrate that the obtained information from the Painlevé test for reductions of nonlinear evolution dissipative differential equations can be used to construct the nonautonomous first integrals of nonlinear ordinary differential equations. Taking into account the found first integrals, we also obtain analytical solutions of nonlinear evolution dissipative differential equations. Our approach is illustrated to obtain the nonautonomous first integrals for reduction of the Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid. The obtained first integrals are used to construct exact solutions of the above-mentioned nonlinear evolution equations with as many arbitrary constants as possible. It is shown that some exact solutions of the equation for description of nonlinear waves in a convecting liquid are expressed via the Painlevé transcendents.