TY  - JOUR
AU  - Farias, Matheus dos Santos
PY  - 2026
DA  - 2026/05/06
TI  - A Proposal Toward the Painlevé Equivalence Problem
JO  - Journal of Mathematics and Interdisciplinary Applications
T2  - Journal of Mathematics and Interdisciplinary Applications
JF  - Journal of Mathematics and Interdisciplinary Applications
VL  - 2
IS  - 2
SP  - 59
EP  - 73
DO  - 10.62762/JMIA.2026.937794
UR  - https://www.icck.org/article/abs/JMIA.2026.937794
KW  - painlevé equivalence problem
KW  - nonlinear ordinary differential equations
KW  - canonical normalization
KW  - differential obstructions
KW  - point equivalence
AB  - This article develops a constructive approach to the Painlevé equivalence problem based on explicit normalization and residual obstruction analysis. Instead of starting from the full invariant complexity of the general point-equivalence problem, we use target-adapted reductions that convert equivalence into a finite algebraic comparison problem inside canonical normalized families. For derivative-free polynomial classes, the method yields exact results. We obtain a constructive criterion for equivalence to the first Painlevé equation in the quadratic case and an exact criterion for equivalence to the second Painlevé equation in the cubic case. In both settings, the method provides explicit recognition and non-equivalence conditions through finite scalar obstructions. For the higher Painlevé equations \(P_{\mathrm{III}}\), \(P_{\mathrm{IV}}\), \(P_{\mathrm V}\), and \(P_{\mathrm{VI}}\), we introduce target-adapted normalized families and finite residual obstruction systems that reduce comparison with the canonical Painlevé models to explicit coefficient conditions. This organizes the six Painlevé equations into a hierarchy of canonical geometries and provides a unified constructive framework for their equivalence analysis. The results establish exact reduction theorems for \(P_{\mathrm I}\) and \(P_{\mathrm{II}}\), and a general normalization-based architecture for the higher Painlevé cases. In this way, the paper advances the Painlevé equivalence problem through explicit mathematical construction rather than purely abstract invariant classification.
SN  - 3070-393X
PB  - Institute of Central Computation and Knowledge
LA  - English
ER  - 

@article{Farias2026A,
  author = {Matheus dos Santos Farias},
  title = {A Proposal Toward the Painlevé Equivalence Problem},
  journal = {Journal of Mathematics and Interdisciplinary Applications},
  year = {2026},
  volume = {2},
  number = {2},
  pages = {59-73},
  doi = {10.62762/JMIA.2026.937794},
  url = {https://www.icck.org/article/abs/JMIA.2026.937794},
  abstract = {This article develops a constructive approach to the Painlevé equivalence problem based on explicit normalization and residual obstruction analysis. Instead of starting from the full invariant complexity of the general point-equivalence problem, we use target-adapted reductions that convert equivalence into a finite algebraic comparison problem inside canonical normalized families. For derivative-free polynomial classes, the method yields exact results. We obtain a constructive criterion for equivalence to the first Painlevé equation in the quadratic case and an exact criterion for equivalence to the second Painlevé equation in the cubic case. In both settings, the method provides explicit recognition and non-equivalence conditions through finite scalar obstructions. For the higher Painlevé equations \(P\_{\mathrm{III}}\), \(P\_{\mathrm{IV}}\), \(P\_{\mathrm V}\), and \(P\_{\mathrm{VI}}\), we introduce target-adapted normalized families and finite residual obstruction systems that reduce comparison with the canonical Painlevé models to explicit coefficient conditions. This organizes the six Painlevé equations into a hierarchy of canonical geometries and provides a unified constructive framework for their equivalence analysis. The results establish exact reduction theorems for \(P\_{\mathrm I}\) and \(P\_{\mathrm{II}}\), and a general normalization-based architecture for the higher Painlevé cases. In this way, the paper advances the Painlevé equivalence problem through explicit mathematical construction rather than purely abstract invariant classification.},
  keywords = {painlevé equivalence problem, nonlinear ordinary differential equations, canonical normalization, differential obstructions, point equivalence},
  issn = {3070-393X},
  publisher = {Institute of Central Computation and Knowledge}
}