Asymptotic behaviour of the third Painlevé transcendents in the space of initial values

Abstract

We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlevé equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each solution is compact and connected. Moreover, we prove that any solution with essential singularity at infinity has an infinite number of poles and zeroes, and similarly at the origin.

36 pages, 6 figures. arXiv admin note: text overlap with arXiv:1703.00100, arXiv:1412.3541