On the connection problem for nonlinear differential equation

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.