WKB analysis of higher order Painlevé equations with a large parameter—Local reduction of 0-parameter solutions for Painlevé hierarchies (PJ)(J=I,II-1 or II-2)

AbstractWe generalize the reduction theorem for 0-parameter solutions of a traditional (i.e., second order) Painlevé equation with a large parameter to those of some higher order Painlevé equation, that is, each member of the Painlevé hierarchies (PJ)(J=I,II-1 or II-2). Thus the scope of applicability of the reduction theorem in [KT1] has been substantially enlarged; only six equations were covered by our previous result, while Theorem 3.2.1 of this paper applies to infinitely many equations

Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painleve equation.Comment: 7 pages, 4 figure