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Asymptotic Expansions of Solutions to the Fifth Painlevé equation

A. D. Bruno, A. V. Parusnikova


Abstract: The article is devoted to the study of the fifth Painlevé equation which has 4 complex parameters $\alpha,\beta, \gamma, \delta$. By methods of Power Geometry we look for asymptotic expansions of solutions to the equation when $x \to \infty$. Ten power expansions with two exponential additions each are obtained when $\alpha \ne 0$. Six of them are over integer powers $x$ (they have been already known) and four are over half-integer powers (they are new).  When $\alpha =0$  we computed 4 one-parameter families of exponential asymptotic forms   $y(x)$ and 3 one-parameter families of complicated expansions $x=x(y)$. All exponential additions, exponential asymptotic forms and complicated expansions have not been known before. Here we improve the method of computation of exponential additions to the power expansions as well.