I want to consider the solutions of the following fourth-order ODE: $$ f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0, \tag{$\ast$}$$ where $a,b$ are complex parameters. It turns out that with a Fourier transformation, we get a first-order equation like for the Airy equation $$ f^{(2)}(t)- tf(t)=0. $$ This is due to the multiplicative factor $t$ which becomes $-id/d\tau$ on the Fourier side, whereas the Fourier transforms of $f^{(4)}(t), f^{(1)}(t)$ are $(i\tau)^4\hat f(\tau), i\tau\hat f(\tau)$. Eventually, we find a first-order differential equation on $\hat f$ with $0$ as a regular singular point, so that we can solve $(\ast)$ explicitly.

My question. The special functions solutions of $(\ast)$ are essentially the inverse Fourier transform (say in the tempered distribution sense) of $\tau^2 e^{i\tau^4}$. Do they have a name? Are they studied systematically somewhere?