multivalued solution of the formal limiting Cauchy problem (set $\epsilon=0$) provide the strong limit of $w_\epsilon(x,t)$ for different $x$ and $t$. However, the choice of sheet requires the solution of a discrete maximization problem parametrized by $x$ and $t$, making the limiting behavior harder to calculate than the weak zero-dispersion limit of the BO equation.

### The semiclassical limit of the free linear Schrödinger equation

In this problem, one considers the equation $$i\epsilon\frac{\partial \psi_\epsilon}{\partial t} +
2\epsilon^2\frac{\partial^2\psi_\epsilon}{\partial x^2}=0,\quad 
x\in\mathbb{R},\quad
t>0
\label{eq:Schroedinger}$$ for small $\epsilon>0$, subject to initial data of WKB form $$\psi_\epsilon(x,0)=A(x)e^{iS(x)/\epsilon}$$ with $A$ and $S$ real-valued and independent of $\epsilon$. For suitable $A$ and $S$, the solution to this problem can be written as an integral $$\psi_\epsilon(x,t)=
\frac{e^{-i\pi/4}}{\sqrt{8\pi\epsilon t}}\int_\mathbb{R}
e^{iI(\xi;x,t)/\epsilon}A(\xi)\,d\xi,\quad t>0,\quad I(\xi;x,t):=S(\xi)
+\frac{(x-\xi)^2}{8t}.$$ The dominant contributions to the solution are calculated via the method of stationary phase (see [@Miller], §5.6), and these come from small neighborhoods of points $\xi$ satisfying $I'(\xi;x,t)=0$, that is, solutions $\xi$ of the implicit equation $\xi=x-4S'(\xi)t$. Evaluating the function $2S'(\cdot)$ on both sides of this equation and making the substitution $u^\mathrm{B}=2S'(\xi)$, one arrives at the equivalent form where $u_0(x):=2S'(x)$. Thus, the branches of the multivalued solution of Burgers’ equation with initial condition $u_0$ correspond to 