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

Abstract:

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 $\mathbb {C}$. 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 out in detail for the Painlevé equation P$_1$, $y''=6y^2+z$, for which we find the Stokes multipliers in closed form and global asymptotics in $\mathbb {C}$ 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.