I wonder whether the following integral of Airy functions can be computed? \begin{equation} F(x,y):=\int_{-\infty}^\infty \int_{-\infty}^\infty Ai(x-u)Ai(y-v) e^{ituv}du dv,\quad t \in \mathbb R. \end{equation} It is the convolution of $Ai(x)Ai(y)$ and $e^{itxy}$. (As $Ai(x)$ satisfies the ODE $z^{\prime\prime}-zx=0$, $F(x,y)$ 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 in a form like $Ai(x+y)Bi(x^2+y^{\frac{1}{2}})$, that is, expressed with usual special functions evaluated at algebraic expressions of $x, y$. ]