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

[SpaceWalrus]

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

 

[hunt_mat]

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

 

[SpaceWalrus]

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?

 

[hunt_mat]

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.

 

[SpaceWalrus]

Either really... Second order with n variables, or n order with 2 variables (or n order with m variables).

 

[hunt_mat]

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.