Abstract:
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.
References