$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$ $\newcommand{\sgn}{\operatorname{sgn}}$ $\newcommand{\rank}{\operatorname{rank}}$
Recall that we consider Burgers equation \begin{align} &u_t+uu_x=\varepsilon u_{xx},\qquad t>0, -\infty < x<\infty \label{eq-9.3.1}\\ &u|_{t=0}=f(x). \label{eq-9.3.2} \end{align} assuming that
Condition 1.
$f(x)$ is a smooth function. and $f'(x)$ has at $x^*$ a non-degenerate negative minimum \begin{gather} f'(x^*)<0, \qquad f''(x ^*)=0, \qquad f'''(x^*) >0. \label{eq-9.3.3} \end{gather} For simplicity we assume that it is an only global minimum.
As $\varepsilon=0$ the proper solution has a shock $x=X(t)$ with $t\ge \bar{t}$, \begin{gather} \bar{t}= -\frac{1}{f'(x^*)}, \label{eq-9.3.4} \end{gather} described by (9.2.7)--(9.2.8) \begin{gather} \frac{dX}{dt}=\frac{1}{2}\bigl(f(z_-(X,t)) + f(z_+(X,t))\bigr), \label{eq-9.3.5}\\ X(0)=\bar{x}:=x^* + \bar{t} f(x^*) \label{eq-9.3.6} \end{gather} where $z_\pm (x,t)$ is a solution of (9.2.9) \begin{gather} x= z +tf(z), \label{eq-9.3.7}\\ z_\pm \gtrless x^* \label{eq-9.3.8} \end{gather} and one can prove that this equation has exactly one solution in $(-\infty,x^*)$ and one solution in $(x^*, \infty)$ as $t> \bar{t}$ while for $0< t \le \bar{t}$ there is just one solution at all.
Further, as $t> \bar{t}$ $u(x,t)=f (z_\pm (x,t))$ for $x\gtrless X(t)$.
Remark 1.
One can prove that \begin{gather} \frac{1}{2}\bigl(u_-(t)-u_+(t)\bigr) \sim -\sqrt{\frac{t-\bar{t}}{2f'''(x^*)}} f'(x^*) \qquad\text{as } t>\bar{t}. \end{gather}
The same construction as in Subsection 9.2.1 works for $\bar{t}+ T < t < T^*$ where $T^* >0$ is a sufficiently small constant and $T>0$ is an arbitrarily small constant. Formally the width of the inner zone is $(t-\bar{t})^{-\frac{1}{2}}\varepsilon $ due to Remark 1.
Without loss of the generality one can assume that $v:= \frac{1}{2}(u_+ + u_-)=0$ (one can reach it by change of variable $x\mapsto v -vt$).
Now scaling $t\mapsto (t-\bar{t})T^{-1}$, $x\mapsto (x-\bar{x}) T^{-\frac{3}{2}}$, $u\mapsto u T^{-\frac{1}{2}}$ we get $\varepsilon \mapsto \varepsilon ' := \varepsilon T^{-\frac{3}{2}}$ and therefore we arrive to
Remark 2.
Therefore we need to cover near shock wave formation \begin{gather} \bar{t} - T < t < \bar{t}+ T \qquad \text{with } \ \qquad T := \varepsilon ^{\frac{2}{3}-\delta}. \label{eq-9.3.11} \end{gather} The width of the inner zone is expected to be $\varepsilon ^{\frac{2}{3}(1-\delta)}$.
So, we get a zone $\{(x,t)\colon |x-\bar{x}| \le \varepsilon ^{\frac{2}{3}-\delta}, |t-\bar{t}| \le \varepsilon ^{\frac{2}{3}-\delta}\}$ which is a inner zone (rank 2) while $\{(x,t)\colon |x-\bar{x}| \le \varepsilon T^{-\frac{1}{2}-\delta}, |t-\bar{t}| \ge \varepsilon ^{\frac{2}{3}-\delta}\}$ is a prvious inner zone and the rest is an outer zone. This is an example of the hierarchy of inner zones.
For more details and for more general equation see