On an Airy function of two variables. II (Q1429102)

The author studies the global behavior of the entire functions on \( \mathbb{C}^2\) defined by the integrals \[ z_C(x,y) = \int_C \exp\left(-t^4/4 + xt^2/2 + yt \right)dt \] where \( C\) is an appropriate path of integration in the \(t\)-plane on which the integrand tends rapidly to \( 0\) as \( t \to \infty\). These functions are called Airy functions of two variables or Pearcey integrals. They satisfy an integrable linear Pfaffian system with no singularity in \( (x, y) \in \mathbb{C}^2\) and that the solutions of this system form a complex vector space of dimension \(3.\) If \( \mathbb{C}^2 \) is compactified by \( \mathbb{P}^1 \times \mathbb{P}^1,\) then the system has irregular singularity along \( x=\infty\) and \( y = \infty.\) So the behavior of solutions at these irregular singular points are of interest. The asymptotic behavior of solutions and Stokes multipliers at \(y=\infty\) have already been computed in the author's previous paper [Nonlinear Anal., Theory Methods Appl. 54 A, 755--772 (2003; Zbl 1040.33001)]. Here, the author gave explicit expressions of asymptotic behavior of solutions and computed the Stokes multipliers at \( x = \infty\). The Airy functions of two variables appear as particular solutions of a nonlinear integrable Hamiltonian system called Garnier system which is a generalization of the second Painlevé equation \( P_{II}\). They also appear as a particular case of the general hypergeometric function indexed by the partition \((5) \) of \( 5\) which is defined on a Zariski open subset of Grassmannian manifold \( Gr(2,5).\) For these connections, see \textit{K. Okamoto} and \textit{H. Kimura} [Q. J. Math., Oxf. II. Ser. 37, 61--80 (1986; Zbl 0597.35114)] and \textit{H. Kimura} and \textit{T. Koitabashi} [Kumamoto J. Math. 9, 13--43 (1996; Zbl 0848.33009)].