Numerical Approach to Obtain Solutions of First and Second Painleve Equations

Abstract

The Painleve' equation and their solutions are used in describing various phenomina in plasma physics, nonlinear waves, quantum field theory, general relativity, nonlinear optics, fibre optics, etc. Paul Painlevé, French mathematician and politician built Poincaré's work in investigating nonlinear second order ordinary differential equations with or without singularities to classify their analytic properties. In the process, he discovered the Painlevé transcendents, written in terms of entire functions, are solution to nonlinear second order ordinary differential equations with the Painlevé property, the solutions are free from movable singularities. This property is a strong indicator of the integrability. Here, we consider about solving the first and second Painlevé equations by using a new computational approach based on Python programs. First, the solutions of first and second Painlevé equations are expressed as Laurent series expansions. Then recurrence relation is obtained for each equation. Finally, solutions are obtained with use of aforementioned recurrence relations and Python codes.