Canonical form of a second-order partial differential equation ($U_{\xi \eta },U_{\xi \xi }+U_{\eta \eta },U_{\xi \xi }$)

For a second-order real linear partial differential equation in two variables, an invertible local change of variables simplifies its principal part. At regular points, the hyperbolic case has mixed-derivative principal part $U_{\xi \eta }$, the elliptic case has principal part $U_{\xi \xi }+U_{\eta \eta }$, and the parabolic case has rank-one principal part $U_{\xi \xi }$, after multiplication by nonzero factors. Lower-order terms and forcing must also be transformed with the chain rule; simplifying only the principal part does not in general give the full canonical equation. In the hyperbolic partial differential equation case the new variables can be chosen constant along the two characteristic families.

For a second-order hyperbolic partial differential equation, the usual reduction to a mixed derivative uses twice continuously differentiable characteristic coordinates. Continuity of the principal coefficients alone does not ensure this regularity, even locally. Consider $R=0$, $S=1/2$ and $T=|x|$ near the origin; the discriminant $S^2-RT$ is the positive constant $1/4$. A characteristic coordinate must satisfyThe two characteristic families have first integrals $x$ and $y-x|x|/2$. Locally, any coordinate transverse to the first family has the form $\eta =h(y-x|x|/2)$ with $h^{\prime }\ne 0$. If this were twice continuously differentiable, its restriction at $x=0$ would make $h$ twice continuously differentiable. But its second $x$ derivative approaches $-h^{\prime }(y)$ from the right and $h^{\prime }(y)$ from the left, a contradiction. Independent characteristic coordinates cannot both choose the other family. Thus no twice continuously differentiable characteristic coordinate system exists near the origin.

With sufficiently smooth coefficients, the two distinct characteristic direction fields do yield smooth local coordinates. If $J$ is their Jacobian determinant and $B$ is the transformed mixed coefficient, then $B^2=J^2(S^2-RT)>0$. Division by $2B$ gives the mixed-derivative canonical form, including all lower-order terms produced by the chain rule. This is a local result; global intersections of characteristic curves can create separate obstructions.

For $\xi =x+e^y$, $\eta =x-e^y$, the Jacobian determinant is $-2e^y$ and the transformed domain is $\xi >\eta$. The chain rule gives $u_{yy}=e^{2y}(U_{\xi \xi }-2U_{\xi \eta }+U_{\eta \eta })+e^y(U_{\xi }-U_{\eta })$, while $u_y=e^y(U_{\xi }-U_{\eta })$. The drift terms cancel, leaving $U_{\xi \eta }=0$. Thus the general solution is $F(x+e^y)+G(x-e^y)$; its characteristic curves are the level sets of the two new coordinates.