On representation of the P-Q pair solution at the singular point neighborhood

TL;DR

This paper derives a closed-form series expansion for solutions of P-Q pairs near singular points, including conservation laws, enhancing understanding of their local behavior in differential systems.

Contribution

It introduces a novel closed-form series expansion for P-Q pair solutions at singular points, along with associated conservation laws, for both regular and irregular singularities.

Findings

Closed-form series expansion for solutions near singularities

Conservation laws derived from compatibility conditions

Applicable to both regular and irregular singularities

Abstract

The compatible expansion in series of solutions of both the equations of P-Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P-Q pair are derived.