Parabolic Monge-Ampère equations in problems of geometry and meteorology (Q677773)

We consider Monge-Ampère equations with unknown function \(z\) depending on two arguments \((x,y)\). For its derivatives \(z_x',\dots\), we use the Monge designations \(p=z_x'\), \(q=z_y'\), \(r=z''_{xx}\), \(s= y''_{xy}\), \(t=z''_{yy}\). We consider the equation \((r+a(y))(t+ b(y))- s^2=0\) and the Cauchy problem for this equation \(z(x,0)=\varphi(x)\), \(q(x,0)=\psi(x)\), \(x\in I\), where \(I\) is an interval, possibly infinite, of the \(x\)-axis. We suggest the exact solution in the partial case when \(\varphi\) and \(\psi\) are quadratic polynomials and discuss applications to geophysics.