Some classification results for hyperbolic equations \(F(x,y,u,u_x,u_y,u_{xx},u_{xy},u_{yy})=0\) (Q1581859)

Let \[ F(x, y, u, p, q, r, s, t)=0\tag{1} \] be a hyperbolic partial differential equation, where \(p = u_x\), \(q=u_y\), \(r=u_{xx}\), \(s=u_{xy}\), \(t=u_{yy}\). The equation (1) is said to be contact equivalent to the equation \[ F(\overline{x},\overline{y},\overline u,\overline u_{\overline x},\overline{u}_{\overline{y}},\overline{u}_{\overline{x}\overline{x}},\overline{u}_{\overline{x}\overline{y}},\overline{u}_{\overline{y}\overline{y}})=0\tag{2} \] if there is an invertible transformation \((x,y,u,u_x,u_y)\to (\overline x,\overline y,\overline u,\overline{u}_{\overline{x}}, \overline{u}_{\overline{y}})\) such that the solutions of (1) are mapped to the solutions of (2). A necessary and sufficient condition for (1) to be contact equivalent to an \(f\)-Gordon equation \[ s +f(x,y,u,p,q) = 0 \] is given in terms of the characteristic vector fields. Necessary and sufficient conditions in order that (1) is contact equivalent to the wave equation \(s = 0\) or to an \(f\)-Gordon which is linear in \(p\) and \(q\) are given in terms of the generalized Laplace invariants. The author also gives necessary and sufficient conditions in order that the equation (1) is contact equivalent to an Euler-Poisson-Darboux equation or Liouville equation or a linear equation or the Klein-Gordon equation \(s = u\). This type of results are established for a hyperbolic Monge-Ampère equation, a Fermi-Ulam-Pasta equation and a number of other equations.