It is known that Painleve II has only simple poles t_n as singularities and these poles have non-movable property, i.e., they depend only on the equation rather than initial conditions. One can try to search for solution as

\begin{equation} y = \sum y_n/(t-t_n), \end{equation}

where the coefficients are determined ( local analysis) but t_n are indeterminate.

Question. Has anybody tried this approach and does it lead to something useful?

By "useful" I mean e.g. asymptotic expansion in n , as dictated by "resurgent philosophy" ( Ecalle, Costin, Sauzin, ... )

\begin{equation} t_n = \sum A_{a,b} n^a log^b(n) \end{equation}

for some $A_{a,b}$ determined e.g. recursively.

It is known that Painlevé II has only simple poles $t_n$ as singularities and these poles have non-movable property, i.e., they depend only on the equation rather than initial conditions. One can try to search for solution as

\begin{equation} y = \sum y_n/(t-t_n), \end{equation}

where the coefficients are determined (local analysis) but $t_n$ are indeterminate.

Question. Has anybody tried this approach and does it lead to something useful?

By "useful" I mean e.g. asymptotic expansion in $n$, as dictated by "resurgent philosophy" (Écalle, Costin, Sauzin, …).

\begin{equation} t_n = \sum A_{a,b} n^a log^b(n) \end{equation}

for some $A_{a,b}$ determined e.g. recursively.