Compatibility of Cauchy data on a characteristic curve

For the canonical hyperbolic partial differential equation $V_{\xi \eta }=0$, every solution is $V=g(\xi )+h(\eta )$. Data on the characteristic $\eta ={\eta }_0$ determine $g$ up to an additive constant, but only determine $h$ at one point. If a transverse derivative is also prescribed, it must satisfy $V_{\eta }(\xi ,{\eta }_0)=h^{\prime }({\eta }_0)$, independent of $\xi$. A varying prescribed value is incompatible; a constant compatible value still leaves the rest of $h$ arbitrary. Thus characteristic Cauchy data can produce nonexistence or nonuniqueness. A derivative in original coordinates must first be transformed by the chain rule before applying this compatibility test.