Tronquée Solutions of the Painlevé Equation PI

Abstract

We analyze the one parameter family of tronquée solutions of the Painlevé equation PI in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter, and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the parameter be zero. We show how this analysis in conjunction with the asymptotic study of the pole sector of the tritronquée in [12] leads to a closed form expression for the Stokes multiplier directly from the Painlevé property, not relying on isomonodromic or related type of results.

A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

“…A natural procedure relying on the Poincaré return map leads to two candidates for asymptotically conserved quantities, see [13],…”

“…Beyond the edges of the sector −π/2 arg x π/2, h develops arrays of poles (unless h is tritronquée). These facts are proved, together with the location of the first few arrays of singularities, in [11] and [13] and are overviewed below.…”

“…General results about the solutions of (3) which decay in some direction at infinity, which correspond to tronquée of (1), are overviewed in [13].…”

“…The existence a Stokes multiplier such that (7) and (8) hold is known in a wide class of differential equations, see in [9] formula following (1.15) and also see in [10], (166). A complete Borel summed expansion of the solutions h satisfying (7), (8) is given in [13]Theorem 2 and (55)-where C + = C, C − = C + µ.…”

“…The tritronquée h t has the sector of analyticity as in (6); h t has an array of poles for arg(x) = −π/2+o(1). For more details see [13].…”