Asymptotics and total integrals of the ${\mathrm{P}}_{\mathrm{I}}^2$ tritronquée solution and its Hamiltonian

Abstract

We study the tritronquée solution $u(x,t)$ of the ${\mathrm{P}}_{\mathrm{I}}^2$ equation, the second member of the Painlevé I hierarchy. This solution is pole-free on the real line and has various applications in mathematical physics. We obtain a full asymptotic expansion of $u(x,t)$ as $x\to \pm \infty$, uniformly for the parameter $t$ in a large interval. Based on this result, we successfully derive the total integrals of $u(x,t)$ and the associated Hamiltonian.

23 pages, 3 figures