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

Abstract

Abstract. We calculate the Stokes multipliers in closed form for P1 using a direct approach. For this purpose, we introduce a new rigorous method, based on Borel summability and asymptotic constants of motion generalizing previous results, to analyze singular behavior of nonlinear ODEs in a neighborhood of infinity and provide global information about their solutions in C. In equations with the Painlevé-Kowalevski (P-K) property (stating that movable singularities are not branched) the method allows for solving connection problems. The analysis is carried in detail for P1, y ′′ = 6y 2 + z, for which we find the Stokes multipliers in closed form and global asymptotics in C for solutions having power-like behavior in some direction, in particular for the tritronquées.Calculating the Stokes multipliers solely relies on the P-K property and does not use linearization techniques such as Riemann-Hilbert or isomonodromic reformulations.We develop methods for finding asymptotic expansions in sectors where solutions have infinitely many singularities. These techniques do not rely on integrability and apply to more general second order ODEs which, after normalization, are asymptotically close to autonomous Hamiltonian systems.

Tronquée Solutions of the Painlevé Equation PI

“…One of the most significant developments in a century of study of Painlevé equations is the isomonodromic approach (and related ones), originating in [29], [20], [31], [27] and further developed [7], [17], [18], [24], [28]. A question is whether a complete asymptotic description and explicit connection formulae are a direct consequence of the Painlevé property, or more structure is needed, e.g., the existence of an isomonodromic representation; this paper together with [12] are a step toward a positive answer to this question. In §3 we show that the transseries representation together with a new method to describe solutions in singular regions [12] allows for a closed form calculation of the Stokes multiplier by direct asymptotic methods using the meromorphicity of solutions (not relying on isomonodromic-type methods).…”

“…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.…”

Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I

“…3. As a first step of the extrapolation into the complex plane, we show that the perturbative large x data in (8) permits a remarkably high-precision extraction of the PI Stokes constant, which is known analytically from isomonodromy methods [20,34] and also from resurgent asymptotics [31,32]. This enables probing of Stokes transitions, and access to higher Riemann sheets, purely from the asymptotic data on the positive real line.…”

“…The agreement with the n → ∞ asymptotics is remarkable, even all the way down to n = 1 for the pole closest to the origin. Asymptotic formulas for successive layers of lines of poles can be derived straightforwardly from recursion relations for adiabatic invariants [31,32].…”

“…Plot of the tritronquée connection formula(32). The blue curve is the real part of the left-hand-side of (32), while the red curve is the real part of the right-hand-side, using our Painlevé I solution y(x) extrapolated into the complex plane.…”