Note that under our assumptions on $u_0'$ we have $T_-<0<T_+$. Also, $T_-$ is the supremum of all $t_\xi<0$ while $T_+$ is the infimum of all $t_\xi>0$. When we consider the Cauchy problem for $t>0$, we will refer to $T:=T_+$ as the *breaking time*.

For $t/t_\xi> 1$ there are *caustic curves* $x^-_\xi(t)<x^+_\xi(t)$ with limiting values as $t\to t_\xi$ given by $x^-_\xi(t_\xi)=x^+_\xi(t_\xi)=x_\xi$ that bound the triply-folded region emerging from $(x_\xi,t_\xi)$. The caustic curves correspond to double roots of , and crossing one of them at a generic point results in a change in the number of sheets by exactly two. Except along the union of the caustic curves and the breaking points from which they emerge, the number of solutions of is always odd, and all are simple roots. See Figure \[fig:CausticsAnnotated\].

![*Except along the caustic curves $x=x^-_\xi(t)$ and $x=x^+_\xi(t)$ the number of solutions of is of the form $2P+1$, and these solutions are simple roots. For this figure, $u_0(x):=2(1+x^2)^{-1}$.*[]{data-label="fig:CausticsAnnotated"}](CausticsAnnotated.pdf)

For the initial data $u_0(x)=2(1+x^2)^{-1}$ used in Figure \[fig:numerics1\], the breaking time before which there is a unique solution for all $x\in\mathbb{R}$ and after which there is an expanding interval in which there are three solutions, is exactly $T=2\sqrt{3}/9\approx
0.3849$. Snapshots of 