How Do We Classify Higher-Order PDEs with Multiple Independent Variables?

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SpaceWalrus
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Suppose you have a PDE with an arbitrary number of independent variables (not necessarily two), and of order n. Is there a nice classification akin to the hyperbolic, parabolic, etc.

Thanks
 
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There is, it's to do with the Monge cone (I think). I also am informed that there are equations which have no classification.
 
I also am informed that there are equations which have no classification.

This surprises me. Is this because forming a general classification system its more complicated than I imagine, or just that doing so serves little to no purpose?
 
It depends if you're saying is there a classification system for second order PDEs in n variables or if there is a classification system for PDEs with order n derivatives.
 
Either really... Second order with n variables, or n order with 2 variables (or n order with m variables).
 
For second order equations in n variables, then it's to do with the Monge cone, with the other case I am not too sure as I am not an expert in this topic.