Asymptotics and total integrals of the \(\mathrm{P}_{\mathrm{I}}^2\) tritronquée solution and its Hamiltonian (Q6598460)

The tritronquée solution of the \(\mathrm{P}^2_{\mathrm{I}}\) equation, the second member of the Painlevé I hierarchy, \N\[\Nu_{xxxx}+10u_x^2 +20 uu_{xx} +40(u^3-6tu +6x)=0\N\]\Nhas various applications in mathematics and mathematical physics, say, Dubrovin's universality conjecture for Hamiltonian PDEs, random matrix theory, and orthogonal polynomials. This paper presents a full asymptotic expansion for the tritronquée solution of the form \N\[ \Nu(x,t)=\frac{z_{\pm}} 2(\pm x)^{1/3} + E(x,\mu), \quad E(x,\mu) \sim |x|^{1/3} \sum_{k=1}^{\infty} \frac {e^{\pm}_k(\mu)} {|x|^{7k/3}}\N\]\Nas \(x \to \pm \infty\). Here (i) \(\mu=t x^{-2/3}\), and each of \(z_{\pm}=z_{\pm}(\mu)\) is the unique real root of \(z^3_{\pm}-24 \mu z_{\pm} \pm 48=0;\) (ii) for any fixed \(M \in (0,2^{-2/3}3^{-1/3})\), \(e^{\pm}_k(\mu)\) with \(k \in \mathbb{N}\) are bounded uniformly for \(\mu \in (-\infty,M]\); and (iii) for every \(k\in\mathbb{N}\), \(e^{\pm}_k(\mu)=O(\mu^{-2})\) as \(\mu \to -\infty\). Furthermore, the authors give total integrals of \(u(x,t)\) and of an associated Hamiltonian \(H_1(x,t)\) vanishing for any fixed \(t\in \mathbb{R}.\) This integral for \(H_1(x,t)\) is written as \N\[\N\int^{+\infty}_{-\infty} \Bigl(H_1(x,t)+\frac 34 6^{1/3}x^{4/3}+3\cdot 6^{-1/3} tx^{2/3} +t^2 +\frac{6^{1/3}}9 t^3x^{-2/3} +\frac{x}{36(x^2+1)} \Bigr) dx=0,\N\]\Nwhich is applied to the asymptotics of the Airy kernel determinant. These results are obtained by the steepest descent analysis for the RH problem related to the P\(^2_{\mathrm{I}}\) equation.