Exponentially-improved asymptotics and numerics for the (un)perturbed first Painlevé equation

The solutions of the perturbed first Painlevé equation \(y"=6y^2-x^\mu\), \(\mu>-4\), are uniquely determined by the free constant \(C\) multiplying the exponentially small terms in the complete large \(x\) asymptotic expansions. Full details are given, including the nonlinear Stokes phenomenon, and the computation of the relevant Stokes multipliers. We derive asymptotic approximations, depending on \(C\), for the locations of the singularities that appear on the boundary of the sectors of validity of these exponentially-improved asymptotic expansions. Several numerical examples illustrate the power of the approximations. For the tri-tronquée solution of the unperturbed first Painlevé equation we give highly accurate numerics for the values at the origin and the locations of the zeros and poles.