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As in case (i), the initial curve is characteristic and its displacement data force $g^{\prime }=1$. The new derivative data would therefore requireThe right-hand side varies along the initial segment. A single number $h^{\prime }(0)$ cannot satisfy those conditions, so no classical solution exists.

Characteristic Cauchy data impose a compatibility condition on transverse derivatives without determining the missing independent wave family. Compatible data can leave nonuniqueness, as in case (i); incompatible data have no solution, as here. The distinction follows from compatibility of Cauchy data on a characteristic curve, rather than a noncharacteristic existence-and-uniqueness theorem.

The initial curve $y^3=9x$ is $\eta =0$, with $\xi =2\sqrt{x}\in [1,2]$, so it is characteristic. The displacement data force $g(\xi )+h(0)=\xi$ and therefore $g^{\prime }=1$ on that interval. Along the curve,The derivative data in case (i) are compatible precisely when $h^{\prime }(0)=-1$. They place no restriction on the rest of $h$. For example, for every real constant $C$,solves the equation and has both prescribed data on the curve. Choosing $C=0$ and $C=1$ exhibits more than one solution directly. The initial curve follows one characteristic family, so its data do not determine the other wave family. This is the compatible case of compatibility of Cauchy data on a characteristic curve.