 Solution

Parametrize the initial curve by $(x_0(r),y_0(r))=(1,r)$, $-1\le r\le 1$. Its tangent is $(0,1)$, whereas the projected characteristic vector field there is $(1,2r+\beta )$. The determinantproves that these are Noncharacteristic Cauchy data: the initial curve is everywhere transverse to the characteristic curves.

There is a regularity qualification. The prescribed function $g(r)=\min \{r,0\}$ is continuous and Lipschitz continuous, but is not differentiable at $r=0$. Thus the data are Cauchy in this geometric sense, with data given by a smooth function on each of $r<0$ and $r>0$; the usual theorem giving a global classical solution from continuously differentiable data does not apply across $r=0$.