Equivalence of the Darboux and Gardner methods for integrating hyperbolic equations in the plane (Q1804683)

Recall that the classical Darboux method can be applied if there exists a pair of functions \(f\), \(g\) which are constant on one of the Monge characteristics; then the intermediate integral \(\varphi (f)= \psi(g)\) together with the original differential equation have a Cauchy characteristic vector field (for arbitrary fixed \(\varphi\), \(\psi\)) and so the problem is reduced to ordinary differential equations. On the other hand, the recent Gardner method deals with a Pfaffian system \({\mathcal S}\) and directly searches for a vector field \(X\) which is Cauchy characteristic for an appropriately extended Pfaffian system \({\mathcal S}' \supset {\mathcal S}\); such \(X\) generates solutions of \({\mathcal S}'\), hence of \({\mathcal S}\). The authors prove that the Gardner method applied to a prolongation of a hyperbolic equation \(F(x,y, u,u_ x, u_ y, u_{xx}, u_{xy}, u_{yy})=0\) is equivalent to the Darboux method for its next prolongation. Lucid exposition is concluded with examples \((x+y) u_{xy}+ n(u_ x+ u_ y)=0\), \(y_{yy}= k(u_ x) u_{xx}\).