= Hyperbolic equation reduced by an exponential characteristic coordinate
{title2=$e^{2y}u_{xx}+u_y=u_{yy}\ \Longrightarrow\ U_{\xi\eta}=0$}
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.
Back to article page
