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Nonlinear hyperbolic equations with characteristic ring of dimension 3. (Russian. English summary) Zbl 1249.35220

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.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B06 Symmetries, invariants, etc. in context of PDEs
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