Paper detail

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

The solutions of the perturbed first Painlevé equation $y"=6y^2-x^μ$, $μ>-4$, 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.