= 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 $V_\eta(\xi,\eta_0)=h'(\eta_0)$, 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.
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