 Cole-Hopf solution for a Burgers N-wave

For the negative-flux viscous Burgers equation, the Cole-Hopf transformation uses $f=2\alpha {\partial }_{\theta }\log \psi$. Initial N-wave data give ${\psi }_0=\exp [U(L^2-{\theta }^2)/(4\alpha )]$ inside $[-L,L]$ and one outside. Completing the square in the heat kernel convolution gives, with $a=1+UZ$,The Gaussian interval masses and their exponentially weighted tails must be treated together when taking the small-diffusion limit.