= Square-root reduction of a quadratic eikonal equation
{title2=$|\nabla u|^2=4u,\quad u>0\implies |\nabla\sqrt u|=1$}
On an open set where $u>0$, put $v=\sqrt u$. The <chain rule> gives $\nabla u=2v\nabla v$, so the quadratic <eikonal equation> is equivalent to $|\nabla v|=1$. For constant positive data $v=v_0$ on a smooth initial curve, the tangential derivative vanishes and the two allowed normal derivatives are $\partial_nv=\pm1$. Locally the two solutions are $v=v_0\pm d$, where $d$ is oriented <distance> along the normal rays; hence $u=(v_0\pm d)^2$ wherever the chosen $v$ stays positive and the normal-coordinate map remains invertible.
There are two separate obstructions to using this initial-data chart. A <characteristic caustic> occurs when normal rays focus and the projection <Jacobian determinant> vanishes. Alternatively $v$ can reach zero while the geometric rays remain separated: the square-root transformation then ceases to apply and the original <Charpit's equations> have a degenerate zero-gradient state. A continuation of $u$ beyond either obstruction needs separate justification; squaring an algebraic expression alone does not preserve a prescribed positive square-root branch.
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