I know how to use Watson's Lemma to find asymptotic expansions for integrals of the form: $$ F_1(\lambda) = \int_0^\infty \phi(t) e^{-\lambda t} dt$$ under suitable conditions on the integrability of $\phi(t)$. An extension of this method can be applied integrals where the exponential term is changed to $F_2(\lambda) := \int_{-a}^a \phi(t) e^{-\lambda t^2} dt$ or even $F_4(\lambda) = \int_{-a}^a \phi(t) e^{-\lambda t^4} dt $. To find the asymptotics of $F_2$ and $F_4$ one essentially splits the integral in odd and even parts, and reduce it to one of type $F_1$, and apply Watson's directly.
However, what can we say about $$ F_3(\lambda) = \int_{\gamma} \phi(t) e^{-\lambda t^3} dt,$$ where $\gamma$ is some contours in the complex plane passing through zero?
For these types of integrals one usually tries to use Laplace method, but since the Inverse Function Theorem fails when applied to the change of variables $t^3 = s^2$, it cannot reduced to an integral of shape $F_2$.
Cheers, VECH