Some Classification Results for Hyperbolic Equations F(x, y, u, ux, uy, uxx, uxy, uyy)=0

We provide a contact invariant characterization for equations of the formuxy+a(x, y, u) ux+b(x, y, u) uy+c(x, y, u)=0,uxy+a(x, y) ux+b(x, y) uy+c(x, y, u)=0,uxy+c(x, y, u)=0,uxy=0. We classify all equations of the form uxy+f(x, y, u, ux, uy)=0 for which the two Ovsiannikovʹs invariants are constants. These results include characterization of the Klein–Gordon equation uxy=u, the Liouville equation uxy=eu, and the class of Euler–Poisson–Darboux equations. It is shown that the wave equation uxy=0, Liouville equation, and the linear equation uxy=2u/(x+y)2 are the only variational equations Darboux integrable at level one. We also show that a hyperbolic Monge–Ampére equation Darboux integrable at level one is equivalent to an equation of type uxy+f(x, y, u, ux, uy)=0. We prove that the hyperbolic Fermi–Ulam–Pasta (FPU) equation uyy=κ(ux)2 uxx is contact equivalent to a linear equation of type uxy=c(x+y) u and we classify all FPU equations Darboux integrable at level one. We also apply our results to equations of type uxy=F(u, ux) that describe pseudo-spherical surfaces.