On the connection problem for nonlinear differential equation

Abstract

We consider the connection problem of the second nonlinear differential equationIn view of that equation (0.1) is equivalent to the fifth Painlevé (PV) equation after a Möbius transformation, we are able to study the connection problem of equation (0.1) 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 (0.1) 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.

On the connection problem for the second Painlevé equation with large initial data

“…We consider the initial problem (1.12) in the two special cases stated in (1.13). The main technique is based on the method of uniform asymptotics introduced by Bassom et al [1], and further applied by [13,14,16,20,21,22,23].…”

On the Connection Problem for Painlevé Differential Equation in View of Geometric Function Theory

“…Bilman et al planned the fundamental rogue wave solutions of PV-III [11]. Newly, Zeng and Hu [12] suggested the connection problem of the second nonlinear differential equation involving a type of PV and they considered the asymptotic expansion solution.…”