On the connection problem for nonlinear differential equation

TL;DR

This paper investigates the connection problem for a specific nonlinear differential equation related to the fifth Painlevé equation, deriving explicit connection formulas using uniform asymptotics, which are relevant for understanding level spacing functions.

Contribution

The paper establishes explicit connection formulas for solutions of a nonlinear differential equation linked to Painlevé V, using uniform asymptotics, providing an alternative to isomonodromy and WKB methods.

Findings

Derived explicit connection formulas relating solution constants to parameter a.

Obtained a monotonically increasing solution on the real axis.

Linked the differential equation to Painlevé V for analysis.

Abstract

We consider the connection problem of the second nonlinear differential equation \begin{equation} \label{eq:1} \Phi''(x)=(\Phi'^2(x)-1)\cot\Phi(x)+ \frac{1}{x}(1-\Phi'(x)) \end{equation} subject to the boundary condition $\Phi(x)=x-ax^2+O(x^3)$ ($a\geq0$) as $x\to0$. In view of that equation (1) is equivalent to the fifth Painlev\'e (PV) equation after a M\"obius transformation, we are able to study the connection problem of equation (1) by investigating the corresponding connection problem of PV. Our research technique is based on the method of uniform asymptotics presented by Bassom el at. The monotonically solution on real axis of equation (1) is obtained, the explicit relation (connection formula) between the constants in the solution and the real number $a$ is also obtained. This connection formulas have been established earlier by Suleimanov via the isomonodromy deformation theory and the WKB method, and recently are applied for studying level spacing functions.