stationary phase points $\xi$ that yield the leading term of the solution $\psi_\epsilon(x,t)$ in the semiclassical limit $\epsilon\downarrow 0$. Unlike in the analysis of Laplace-type integrals, where only the critical points corresponding to maxima matter in the limit, for oscillatory integrals all stationary phase points contribute to the leading-order behavior, and therefore we have an asymptotic representation of $\psi_\epsilon(x,t)$ as a sum over branches $u^\mathrm{B}_n(x,t)$ of the multivalued solution of Burgers’ equation with initial data $u_0$: $$\psi_\epsilon(x,t)=\sum_{n=0}^{2P(x,t)}M_n(x,t)e^{i\theta_k(x,t;\epsilon)} + O(\epsilon),\quad t>0,
\label{eq:psiSP}$$ where $M_n(x,t)$ are slowly-varying positive amplitudes given by $$M_n(x,t):=A(x-2u_n^\mathrm{B}(x,t)t)\sqrt{\left|1-2t\frac{\partial 
u_{n}^\mathrm{B}}{\partial x}(x,t)\right|},$$ and $\theta_k(x,t;\epsilon)$ are rapidly-varying real phases given by $$\theta_n(x,t;\epsilon):=\frac{1}{\epsilon}I(x-2u_n^\mathrm{B}(x,t)t;x,t) + 
\frac{\pi}{4}
\left(\mathrm{sgn}\left(1-2t\frac{\partial u_{n}^\mathrm{B}}{\partial x}(x,t)
\right)-1\right),$$ for $n=0,\dots,2P(x,t)$.

A more explicit connection with the multivalued solution of Burgers’ equation may be obtained by introducing the quantity $$w_\epsilon(x,t):=2\epsilon\frac{\partial}{\partial x}
\Im\{\log(\psi_\epsilon(x,t))\}$$ which is the fluid velocity in Madelung’s interpretation of the wave function $\psi_\epsilon$ as describing a quantum-corrected fluid motion. Under the condition that the error term in becomes $O(1)$ after differentiation with respect to $x$, some easy calculations show that implies $$w_\epsilon(x,t)=\Re\left\{
\frac{\displaystyle\sum_{n=0}^{2P(x,t)}u_n^B(x,t)M_n(x,t)e^{i\theta_n(x,t;\epsilon)}}
{\displaystyle\sum_{n=0}^{2P(x,t)}M_n(x,t)e^{i\theta_n(x,t;\epsilon)}}\right\}
+O(\epsilon).$$ It is then easy to see that if $P(x,t)=0$, $w_\epsilon(x,t)$ converges strongly pointwise to $u_0^B(x,t)$, the unique solution (for this $x$ and $t$, anyway) of 