= Solution
Set $s=Z+c\theta$ and write $q=Q(s)$. Substitution into the <modified Burgers equation with cubic flux> gives
$$
(1-cQ^2)Q'=\epsilon c^2Q''.
$$
After one integration,
$$
\epsilon c^2Q'=Q-\frac c3Q^3+C.
$$
A regular constant limiting state has $Q'\to0$. The state $Q=0$ at the negative end sets $C=0$, and the nonzero state $Q=\beta$ at the positive end gives
$$
\boxed{c=\frac3{\beta^2}.}
$$
Thus the first-order equation is $\epsilon c^2Q'=Q(1-Q^2/\beta^2)$. Put $W=Q^2$. Its equation is logistic,
$$
W'=\frac2{\epsilon c^2}W(1-W/\beta^2).
$$
The heteroclinic <travelling wave> connecting the prescribed endpoints is consequently
$$
\boxed{q(\theta,Z)=\frac{\beta}{\sqrt{1+\exp[-2(Z+c\theta-s_0)/(\epsilon c^2)]}},\qquad c=3/\beta^2,}
$$
where $s_0$ is an arbitrary translation. Taking the sign from $\beta$ is essential; squaring the equation alone loses the negative-front solution. A fixed level propagates with $d\theta/dZ=-1/c=-\beta^2/3$, also obtained from the <Rankine-Hugoniot condition> for flux $-q^3/3$.
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