Has any one computed the following integral of airy functions? \begin{equation} F(x,y):=\int_{-\infty}^\infty \int_{-\infty}^\infty Ai(x-u)Ai(y-v) e^{iuv}du dv,\quad u,v\in \mathbb R. \end{equation} It is the convolution of $Ai(x)Ai(y)$ and $e^{ixy}$. I could not recognize whether it can be identified with any usual special functions. (As $Ai(x)$ satisfies the ODE $x^{\prime\prime}-tx=0$, $F(x,y)$ might be the solution of a good PDE; identifying the PDE and its general solutions may help with solving the integral explicitly.)

[An ideal answer would be of the form like $Ai(x+y)Bi(x^2+y^{\frac{1}{2}})$, that is, expressed with usual special functions evaluated at algebraic expressions of $x, y$. ]