Here we set out algorithm based on the three-dimensional power geometry, which allows to find elliptic asymptotic forms of solutions to a wide class of differential equations, including Painlev'e equations. We use this algorithm and methods of the plane and three-dimensional power geometry to calculate power, complicated, exotic and elliptic asymptotic forms of the solutions to the first four Painlev\'e equations and also exponentially small additions to these asymptotic forms. Asymptotic forms of the solutions to the first Painlev'e equations, expressed in elliptic functions, were found by Boutroux in 1913.

Mathematical problems and theory of numerical methods