On the connection problem for nonlinear differential equation

Abstract

Abstract We consider the connection problem of the second nonlinear differential equation 1 Φ″(x)=(Φ′2(x)−1)cotΦ(x)+1x(1−Φ′(x)) $$\begin{aligned} \varPhi''(x)= \bigl( \varPhi^{\prime2}(x)-1 \bigr)\cot\varPhi(x)+\frac{1}{x} \bigl(1- \varPhi'(x) \bigr) \end{aligned}$$ subject to the boundary condition Φ(x)=x−ax2+O(x3) $\varPhi(x)=x-ax^{2}+O(x^{3})$ as x→0 $x\rightarrow0$. In view of the fact that equation (1) is equivalent to the fifth Painlevé (PV) equation after a Möbius 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 et al. The asymptotic behavior of the monotonic solution as x→∞ $x\rightarrow \infty$ on the real axis of equation (1) is obtained, the explicit relation (connection formula) between the constants appearing in the asymptotic behavior and the real number a are also obtained.

Keywords

• Connection formulas
• Uniform asymptotics
• Painlevé V equation
• Parabolic cylinder function