I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: $e^{-\alpha t}(Acos(\omega_d t) + Bsin(\omega_d t))$
I am interested in a solution of the form $e^{-\alpha t^2}Acos(\omega_d t)$ i.e., I want the oscillations to die at quadratic rate.
Is there a corresponding differential equation that can generate this kind of behavior?
The 3 scenarios of oscillation that you mentioned only show up for 2nd order linear differential equations with constant coefficients. Second order linear differential equations with non-constant coefficients can have a much broader variety of solutions.
As an example of my comment, if we take the derivative of your proposed solution, we get: $$ \frac{d}{dt}\Big( e^{-\alpha t^2} A \cos{\omega_d t} \Big) = -A e^{-\alpha t^2} (2 \alpha t \cos{\omega_d t} + \omega_d \sin{\omega_d t}) = A e^{-\alpha t^2}\cos{\omega_d t}(2 \alpha t + \omega_d \tan{\omega_d t}) $$
So, your function is a solution to the first order linear differential equation
$$ \frac{d}{dt}f(t) = (2\alpha t+\omega_d \tan{\omega_d t}) f(t) $$ Subject to any initial condition that you like to determine the coeficient $A$. We could do a similar thing to find a 2nd, 3rd, etc. order equation.