Nonlinear hyperbolic equations with characteristic ring of dimension 3 (Q2900939)

The paper provides a method of classification of Darboux integrable nonlinear hyperbolic equations \(u_{xy}=f(u,u_{x},u_{y})\) based on investigation of the characteristic pairs of Lie rings. Constructive conditions on the right-hand side \(f\) of the equation with characteristic ring of dimension three are obtained. These equations possess second-order integrals. In particular, a list of equations satisfying the constructive conditions is given for the equation \(u_{xy}=\varphi(u)\psi(u_{x})h(u_{y})\). Formulas for \(x\) - and \(y\)-integrals are given for these equations.