Hyperbolic equation reduced by an exponential characteristic coordinate ($e^{2y}{u}_{xx}+u_y=u_{yy}⟹U_{\xi \eta }=0$)

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.