ASYMPTOTICS AND TOTAL INTEGRALS OF THE PI² TRITRONQUÉE SOLUTION AND ITS HAMILTONIAN.

We study the tritronquée solution u(x, t) of the PI² 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. [ABSTRACT FROM AUTHOR]