Let be $$ \partial_t q + \partial_x (q^2 /2) = \epsilon \partial_{xx}q$$
$$ \begin{equation} q_0(x) = q(x, 0) = \begin{cases} q_l & x<0 \\ q_r & x \ge 0 \end{cases} \end{equation} $$
the viscous Burgers' equation.
By inserting it's easy to verify that the "traviling wave" solution $q_{\epsilon} (x,t) :=w((x-st)/ \epsilon) = w(\xi)$ solves the problem for $q_l > q_r$ (!) with abbreviations
$\xi := (x-st)/ \epsilon$,
$s := (q^2_l /2 - q^2_r /2)/(q_l -q_r)$ and
$w(\xi) := q_r + 1/2(q_l - q_r)(1-tanh((q_l + q_r)\xi/4))$
My question is why do there not exist "traveling wave" solutions for $q_l < q_r$ ?