We consider two special cases of the connection problem for the homogenous second Painlevé equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give an asymptotic classification of the real solutions of PII on the negative (positive) real axis with respect to their initial data which matches the numerical results by Fornberg and Weideman. As by product, a rigorous proof of a property associated with the nonlinear eigenvalue problem of PII on the real axis, recently revealed by Bender and Komijani, is given by deriving the asymptotic behavior of the Stokes multipliers. Some further problems are also to be discussed in this talk. This is a joint work with Zhao-Yun Zeng.

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