Asymptotics and total integrals of the $\mathrm{P}_{\rm I}^{2}$ tritronqu\'{e}e solution and its Hamiltonian

TL;DR

This paper provides a comprehensive asymptotic analysis of the pole-free tritronque9e solution of the d7P_{ m I}^{2} equation, deriving its total integrals and Hamiltonian, with implications for mathematical physics.

Contribution

It presents the first full asymptotic expansion of the d7P_{ m I}^{2} tritronque9e solution and computes its total integrals and Hamiltonian.

Findings

Asymptotic expansion of the solution as x a7 d7 o a7 d7 a7 a0

Explicit formulas for total integrals of the solution and Hamiltonian

Pole-free behavior of the solution on the real line

Abstract

We study the tritronqu\'{e}e solution $u(x,t)$ of the $\mathrm{P}_{\rm I}^{2}$ equation, the second member of the Painlev\'{e} 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.