= Sinusoidal Cole-Hopf solution for negative-flux Burgers flow
{title2=$\psi=I_0(a)+2\sum_{n\ge1}(-1)^nI_n(a)e^{-n^2\delta Z}\cos n\theta,\quad a=1/(2\delta)$}
For $q_Z-qq_\theta=\delta q_{\theta\theta}$ and initial data $q=\sin\theta$, the <Cole-Hopf transformation> $q=2\delta\psi_\theta/\psi$ gives $\psi(\theta,0)=\exp[-\cos\theta/(2\delta)]$. The <Fourier series> of this exponential and the <heat equation> give the displayed exact positive diffusion solution. Its first harmonic gives $q\sim4\delta[I_1(a)/I_0(a)]e^{-\delta Z}\sin\theta$. The <large-argument asymptotic expansion of a modified Bessel function> makes the ratio tend to one as $\delta\to0$.
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