On a certain class of hyperbolic equations with second-order integrals (Q2214379)

In this paper the authors examine a special class of nonlinear hyperbolic equations for a function \(u=u(x,y)\) depending on parameter functions \(p=p(x,y)\) and \(\varphi=\varphi(x,y,u,u_y)\), and possessing a second-order \(y\)-integral \(\bar W = \bar W (x,y,u,u_y,u_{yy})\). They derive a system of nonlinear partial differential equations satisfied by \(\bar W\) and study its solutions under certain simplifying assumptions. The authors provide an explicit form of the \(y\)-integral \(\bar W\), and they describe an explicit procedure for determining the parameter functions \(p\) and \(\varphi\) such that the resulting equation for \(u\) has the \(y\)-integral \(\bar W\). The authors also discuss the structure of \(x\)-integrals of this equation. In the last section they show that the Lane equation is a special case of the hyperbolic equation for \(u\) and obtain the \(x\) and \(y\)-integrals for the Lane equation. The paper presents a topic which will be of interest to specialists in the field. However, it would be of interest to a wider audience if the authors had given a clear motivation for studying the subject and explained in more detail the significance of the equation under consideration.