TY - JOUR

T1 - On singularities of certain non-linear second-order ordinary differential equations

AU - Filipuk, Galina

AU - Kecker, Thomas

N1 - Funding Information:
GF acknowledges the support of the National Science Center (Poland) through the Grant OPUS 2017/25/B/BST1/00931.
Publisher Copyright:
© 2021, The Author(s).

PY - 2022/2/1

Y1 - 2022/2/1

N2 - The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations is applied to obtain information about the singularity structure of the solutions of the corresponding non-linear differential equations. We first deal with the so-called Painlevé example, which passes the Painlevé test, but the solutions have more complicated singularities. Resolving base points in the equivalent system of equations we can explain the complicated structure of singularities of the original equation. The Smith example has a solution with non-isolated singularity, which is an accumulation point of algebraic singularities. Smith’s equation can be written as a system in two ways. We show that the sequence of blow-ups for both systems can be infinite. Another example that we consider is the Painlevé-Ince equation. When the usual Painlevé analysis is applied, it possesses both positive and negative resonances. We show that for three equivalent systems there is an infinite sequence of blow-ups and another one that terminates, which further gives a Laurent expansion of the solution around a movable pole. Moreover, for one system it is even possible to obtain the general solution after a sequence of blow-ups.

AB - The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations is applied to obtain information about the singularity structure of the solutions of the corresponding non-linear differential equations. We first deal with the so-called Painlevé example, which passes the Painlevé test, but the solutions have more complicated singularities. Resolving base points in the equivalent system of equations we can explain the complicated structure of singularities of the original equation. The Smith example has a solution with non-isolated singularity, which is an accumulation point of algebraic singularities. Smith’s equation can be written as a system in two ways. We show that the sequence of blow-ups for both systems can be infinite. Another example that we consider is the Painlevé-Ince equation. When the usual Painlevé analysis is applied, it possesses both positive and negative resonances. We show that for three equivalent systems there is an infinite sequence of blow-ups and another one that terminates, which further gives a Laurent expansion of the solution around a movable pole. Moreover, for one system it is even possible to obtain the general solution after a sequence of blow-ups.

UR - http://www.scopus.com/inward/record.url?scp=85122086034&partnerID=8YFLogxK

U2 - 10.1007/s00025-021-01577-1

DO - 10.1007/s00025-021-01577-1

M3 - Article

AN - SCOPUS:85122086034

VL - 77

JO - Results in Mathematics

JF - Results in Mathematics

SN - 1422-6383

IS - 1

M1 - 41

ER -