 Sinusoidal Cole-Hopf solution for negative-flux Burgers flow

For $q_Z-q{q}_{\theta }=\delta q_{\theta \theta }$ 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 $q\sim 4\delta [I_1(a)/I_0(a)]e^{-\delta Z}\sin \theta$. The large-argument asymptotic expansion of a modified Bessel function makes the ratio tend to one as $\delta \to 0$.