Expansions of solutions to the fifth Painlev´e
equation near its nonsingular point

Abstract:

The article is devoted to the study of the fifth Painlev´e equation which has 4 complex parameters. By methods of Power Geometry we look for asymptotic
expansions of solutions to the equation near its nonsingular point z=z_{0}, z_{0}≠0, z_{0}≠∞ for all values of parameters of the equation. We have proved that there exist exactly 10 families of expansions. These families are power series in the local variable z − z_{0}. One of them is new: it has an arbitrary coefficient of
the (z − z_{0})^{4}. One of these families is two-parameter, other are one-parameter.
All the expansions converge near the point z=z_{0}.

Publication language:russian,
pages:16

Research direction:

Mathematical problems and theory of numerical methods