The original question comes from Exercise #2, section 4.5, "An Introduction to Nonlinear Partial Differential Equations" 2nd Ed by J. David Logan. Please forgive my blatant ask for help but I have no ideas how to start from reading the book.
Consider the scalar equation \begin{equation} u_t + (\phi(u))_x = \mu u_{xx}, \end{equation} where $t, x$ and $u$ are dimensionless variables of order 1, and $\mu = \mu_1 \epsilon^2 + O(\epsilon^3), \epsilon \ll 1$, and $\phi(u)$ can be expanded in a Taylor series about $u = u_0 \in \mathbb{R}$. Now let $x = X(t)$ be a representative location on a wave propagating into the uniform state $u_0$. If we denote $\xi = (x - X(t))/\epsilon$ and assume \begin{equation} u(\xi, t) = u_0 + \epsilon u_1(\xi, t) + \epsilon^2 u_2(\xi, t) + O(\epsilon^3); D(t) = X'(t) = D_0(t) + \epsilon D_1(t) + O(\epsilon^2), \end{equation} show that $u_1$ satisfies the Burgers-like equation $(u_1)_{\tau} + (0.5\alpha (u_1)^2)_{\eta} = \mu_1 (u_1)_{\eta\eta}$, where $\eta$ and $\tau$ are suitable spatial and time variables, respectively, and $\alpha$ is a constant.
Thanks in advance everyone who are willing to help me.