ASYMPTOTICS AND TOTAL INTEGRALS OF THE P²_I TRITRONQUÉE SOLUTION AND ITS HAMILTONIAN

We study the tritronquée solution 𝑢⁡(𝑥, 𝑡) of the P²_I equation, the second member of the Painlevé I hierarchy. This particular solution is also known as the Gurevich–Pitaevskii solution of the KdV equation. It 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 → ±∞, 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 with respect to 𝑥 ∈ ℝ. Surprisingly, although 𝑢⁡(𝑥, 𝑡) exhibits significant differences between 𝑡 > 0 and 𝑡 < 0, both integrals equal zero for all 𝑡. © 2024 Society for Industrial and Applied Mathematics.