Fay solution ($q=2\delta {\sum }_{n\ge 1}\sin (n\theta )/\sinh (n\delta Z)$)

The Fay solution is a periodic solution of the negative-flux viscous Burgers equation for $Z>0$. Its Fourier series has exponentially decreasing coefficients at each positive $\delta Z$. Expanding $1/\sinh (ns)=2{\sum }_{j\ge 0}{e}^{-(2j+1)ns}$ and summing the sine series givesFor $s\gg 1$, the first harmonic gives $q\sim 4\delta e^{-s}\sin \theta$. The Fay shock-layer asymptotics describe its opposite, small-$s$ regime.

For $|\theta |\ll 1$ and $s=\delta Z\ll 1$, the Fay solution becomes $q\sim 4\delta \theta {\sum }_{j\ge 0}[((2j+1)s{)}^2+{\theta }^2{]}^{-1}$. The partial-fraction identity for the hyperbolic tangent sums this to the displayed smooth shock layer. It resolves the odd transition from $-\pi /Z$ to $+\pi /Z$ on angular thickness $O(\delta Z)$; at $\theta =0$ both sides vanish and the comparison is read through their derivatives.