= Burgers characteristic breaking
{c}
{title2=$x=\xi+t u_0(\xi),\quad u_x=\frac{u_0'(\xi)}{1+t u_0'(\xi)}$}
Before <characteristic crossing>, the <method of characteristics> for the <Inviscid Burgers equation> transports each initial value without changing it:
$$
\frac{dx}{dt}=u,\qquad \frac{du}{dt}=0,\qquad x=\xi+t u_0(\xi),\qquad u(x,t)=u_0(\xi).
$$
Here $\xi$ labels the starting position. Suppose $u_0$ is a bounded <continuously differentiable function> with bounded <derivative>. As long as $1+t u_0'(\xi)>0$ for every label, the <characteristic flow map> is invertible and differentiating it gives the displayed formula for $u_x$.<Partial Differential Equations by Victor Ivrii/First order PDEs>[§2.1, Quasilinear equations, Example 7, (2.1.8)–(2.1.9)]{ref}
If the negative minimum $m=\min_\xi u_0'(\xi)$ is attained, the first breaking time is $t_*=-1/m$. Two labels $\xi_1<\xi_2$ with $u_0(\xi_1)>u_0(\xi_2)$ have formal intersection time
$$
t=\frac{\xi_2-\xi_1}{u_0(\xi_1)-u_0(\xi_2)}>0.
$$
Their neighboring-label limit explains why the most negative initial slope breaks first. As $t\uparrow t_*$, the amplitude remains bounded but $u_x$ diverges: this is <gradient blow-up>. Continuing intersecting lines would assign incompatible values at one position; the post-breaking <entropy solution> instead uses <shock waves>. Nondecreasing data have no forward <characteristic crossing> by this mechanism.
For a concrete calculation in <dimensionless quantities> on the infinite line, choose $u_0(\xi)=1-\tanh\xi$, with limiting states $2$ and $0$. Then $u_0'(\xi)=-(1-\tanh^2\xi)$ has minimum $-1$ at $\xi=0$, so $t_*=1$ and the breaking point is $(x,t)=(1,1)$. Along the central <characteristic curve> $x=t$, $u=1$ and $u_x=-1/(1-t)$ before breaking. The labels $\xi=\pm1$ give intersecting formal lines at $t=1/\tanh1\approx1.313$, after the <classical solution> has already broken.
\Image[/burgers-finite-time-characteristic-crossing.png]
{title=Decreasing initial data steepen: the first gradient blow-up is at time 1, before two selected characteristics intersect}
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