= Solution
The solution is constant on <characteristics> $x_1+x_2=\eta$. On the initial curve this invariant is $h(s)=s$ for $s<0$ and $h(s)=s+s^2$ for $s\ge0$. This is a $C^1$ bijection of $\mathbb R$, with everywhere positive derivative. Its inverse is
$$
h^{-1}(\eta)=\begin{cases}\eta,&\eta<0,\\(\sqrt{1+4\eta}-1)/2,&\eta\ge0.\end{cases}
$$
Therefore $\boxed{u(x_1,x_2)=f(h^{-1}(x_1+x_2)).}$ The inverse is $C^1$ even at zero, so for $C^1$ data this is a global $C^1$ solution. Every characteristic meets the initial curve exactly once.
Back to article page
