As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ should capture some behaviour that occurs between the 'macroscopic' classical scale and 'microscopic' quantum scale.
A famous PDE that depends on a small parameter $\epsilon > 0$ is the Ginzburg–Landau equation: \begin{equation} \epsilon \Delta u + \epsilon^{-1} u ( 1 - \lvert u \rvert^2) = 0, \end{equation} defined for $\mathbf{C}$-valued functions $u$. Here one often considers sequences of solutions $(u_{\epsilon_j} \mid j \in \mathbf{N})$ along some sequence $\epsilon_j \to 0$. In the limit as $j \to \infty$ the $u_{\epsilon_j}$ converge weakly away from a codimension two set where the energies concentrate and vortices appear.
Question. Do semiclassical methods play a role in thes analysis of this PDE? In theory, what sort of mathematical questions could be approached from this 'quantum' point of view?
