WKB analysis of higher order Painlevé equations with a large parameter. II. Structure theorem for instanton-type solutions of \((P_J)_m (J= I, 34\), II-2 or IV) near a simple \(P\)-turning point of the first kind (Q533371)

This paper continues a series of papers [the authors, Proc. Japan Acad., Ser. A 80, No.~5, 53--56 (2004; Zbl 1067.34088); Adv. Math. 203, No.~2, 636--672 (2006; Zbl 1108.34065)] on the exact WKB analysis of the instanton-type solutions of the ODEs \((P_J)_m\) with a large parameter which belong to the hierarchies associated with the Painlevé equations PI, PII and PIV as their first members. In one of their previous papers, the authors proved that any \(0\)-parameter instanton-type solution of equation \((P_J)_m\) can be formally and locally transformed near a simple \(P\)-turning point (the turning point of the Fréchet derivative of the nonlinear ODE) of the first kind to a \(0\)-parameter solution of equation PI with a large parameter, \((\lambda_I)_{tt}=\eta^2(6\lambda_I^2+t)\). In the present paper, the authors generalize this result for \((2m)\)-parameter instanton-type solutions of equation \((P_J)_m\). Namely, let the point \(t=\tau\) correspond to a triple turning point of the underlying linear Schrödinger equation. Let the point \(t=\sigma\neq\tau\) be sufficiently close to \(\tau\), belong to a \(P\)-Stokes curve emanating from \(\tau\), and be characterized by the property that the double and single turning points of the associated linear Schrödinger equation are connected by a Stokes line. Then the above mentioned \((2m)\)-parameter instanton-type solution of \((P_J)_m\) can be locally transformed near \(t=\sigma\) to an appropriate \(2\)-parameter instanton-type solution of equation PI.