= Fold caustic for a cubic Charpit equation
{c}
{title2=$xu_x+u_y^3+u=1$}
For initial data $u(x,0)=1+x^3/4$ on $0\leq x\leq1$, the <characteristic> surface has $x=se^r$, $y=3s^2(1-e^{-2r})/2$, $u=1+s^3(e^{-3r}-3e^{-r}/4)$. Its projection determinant is $3s^2e^r(2e^{-2r}-1)$, so the sheet connected to the initial data loses local invertibility at $r=\log2/2$. The regular sheet has $r<\log2/2$ and projects to $0<x<\sqrt2$, $3(1-x^{-2})/2<y<3x^2/8$. Its upper boundary is a <fold point of a critical manifold> <caustic>. Solving the inverse yields $s^2=[x^2+\sqrt{x^4-(8/3)x^2y}]/2$; the other square-root branch lies on the folded sheet and cannot simply be joined as a unique smooth solution. The initial point $s=0$ is already <characteristic>, illustrating the need to inspect endpoints as well as the interior of a Cauchy curve.
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