I've come across the following equation in my work: $$ u_{tt} = \det(\textrm{Hess } u), $$ where $u=u(t,x,y)$ is a function of 3 variables and
$$ \textrm{Hess } u= \left[ {\begin{array}{cc} u_{tt} & u_{tx} & u_{ty}\\ u_{xt} & u_{xx} & u_{xy} \\ u_{yt} & u_{yx} & u_{yy} \end{array} } \right]. $$
I have not been able to find much on this equation (I know there's a big literature on $\det(\textrm{Hess }u)= f$ though) so hoping someone can tell me something/anything about this equation or give me references.
thanks