= Cole-Hopf solution for a Burgers N-wave
{c}
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 $a=1+UZ$,
$$
\psi=1-I_\alpha(\theta,L,Z)+I_\alpha(\theta,La,Za)a^{-1/2}e^{U(L^2-\theta^2/a)/(4\alpha)}.
$$
The <Gaussian interval masses> and their exponentially weighted tails must be treated together when taking the small-diffusion limit.
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