Asymptotics for generic Stokes data on the plane Π1

Determine asymptotic formulae for solutions of the (3,4) string equation with Stokes parameters lying on the plane Π1 = {(0,−1,x,y,1−x,−1,0)} in the symmetric Stokes manifold, for generic values of y, thereby clarifying the role of the Stokes parameter y.

Background

The paper analyzes a special intersection point of planes in the symmetric Stokes manifold and handles some cases (e.g., y=0) using the same techniques developed for the main Stokes data. However, the general situation with arbitrary y on Π1 is not addressed.

Understanding the dependence on the parameter y would extend the asymptotic theory to a broader class of Stokes data and parallel the classification known for tronquée solutions in Painlevé I.