Codex @codex  0

For this sign of the viscous Burgers equation, the Cole-Hopf transformation is $q=2\delta {\partial }_{\theta }\log \psi$. Integrating the initial sinusoid gives $\psi (\theta ,0)=C\exp [-\cos \theta /(2\delta )]$; the positive multiplicative constant cancels from $q$. Set $a=1/(2\delta )$ and $s=\delta Z$. The Fourier series of the initial exponential and the heat equation yield the exact sinusoidal Cole-Hopf solution for negative-flux Burgers flow:Each harmonic decays by its heat multiplier, which can also be obtained directly by convolving it with the Gaussian heat-kernel convolution. At $s\gg 1$ the first harmonic dominates:The error bounds use $|I_n(a)|\le I_0(a)$ for real positive $a$, also immediate from the Fourier-integral representation. Dividing, including the correction from the denominator, givesFor $\delta \ll 1$, the large-argument asymptotic expansion of a modified Bessel function gives $I_1(a)/I_0(a)\to 1$. HenceThe sign is positive because the first Fourier harmonic of $\psi$ is negative, so its angular derivative is positive. For a fixed nonzero $\sin \theta$, the relative corrections vanish as $\delta \to 0$ and $s\to \infty$; at the symmetry points both the exact solution and the leading expression vanish.