Darboux integrable nonlinear hyperbolic equations (Q1363885)

We study equations with two independent variables of the form \[ u_{xy}=F(x,y,u,u_x,u_y).\tag{1} \] It is well-known that there exist two types of integrable equations (1). The first type includes the wave equation, the Liouville equation \(u_{xy}=\exp(u)\), and numerous analogs of the latter. The best-known equation of the second type is the sine-Gordon equation \(u_{xy}=\sin(u)\). Equations of the first type are called Darboux integrable equations, because Darboux appears to be the first to have defined them rigorously and to have suggested a geometric method for their integration. The problem of a complete classification of such equations remains unsolved. We establish a connection between Darboux integrable equations (1) and differentiable substitutions of the type of the Miura transform for evolutionary equations; we suggest an effective criterion of the integrability according to Darboux in terms of the Laplace invariants for the linearization operator and indicate an explicit formula for higher symmetries of Darboux integrable equations (1).