points $\xi=\xi(x,t)\in\mathbb{R}$ at which $R(\xi;x,t)$ achieves its maximum value. The critical points of $R$ satisfy $\xi=x-2u_0(\xi)t$. Writing $u^{\mathrm{B}}=u_0(\xi)$ and applying $u_0$ to both sides gives the equation , so the critical points correspond to the sheets of the multivalued solution of the (inviscid) Burgers equation with initial data $u_0$. It is easy to check that if $x$ and $t$ are such that there is just one sheet, then the unique critical point is the global maximizer of $R$ and Laplace’s method gives the result that $w_\epsilon(x,t)$ converges (strongly, pointwise in $x$ and $t$) to $u^\mathrm{B}(x,t)$. On the other hand, if there are $2P+1>1$ sheets, then for generic $(x,t)$ exactly one of them corresponds to the global maximum of $R$, and Laplace’s method predicts that $w_\epsilon(x,t)$ will converge to the maximizing sheet. Shocks appear in the small viscosity limit as curves in the $(x,t)$-plane along which there are jump discontinuities of the pointwise limit corresponding to sudden changes in the choice of sheet that maximizes the exponent $R$. To summarize, we have the formula $$\lim_{\epsilon\downarrow 0}w_\epsilon(x,t)=u^\mathrm{B}_n(x,t),\quad
n=\mathop{\mathrm{argmax}}_{m=0,\dots,2P(x,t)+1} R(x-2u^\mathrm{B}_m(x,t)t;x,t)$$ for $(x,t)$ not on a shock.

Thus, one sees that for the zero-viscosity limit of the viscous Burgers equation, different sheets of the 