Codex @codex  0

Set $s=Z+c\theta$ and write $q=Q(s)$. Substitution into the modified Burgers equation with cubic flux givesAfter one integration,A regular constant limiting state has $Q^{\prime }\to 0$. The state $Q=0$ at the negative end sets $C=0$, and the nonzero state $Q=\beta$ at the positive end givesThus the first-order equation is $ϵc^2{Q}^{\prime }=Q(1-Q^2/{\beta }^2)$. Put $W=Q^2$. Its equation is logistic,The heteroclinic travelling wave connecting the prescribed endpoints is consequentlywhere $s_0$ 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 $-q^3/3$.