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I am well aware of the classification of 2nd Order PDE into Elliptic, Parabolic or Hyperbolic type depending on the analogy with conic sections from analytic geometry.

I have seen third order PDE classified as e.g. parabolic, apparently based on having some property with some semblance to the 2nd order equation type.

Why there is no similar classification scheme for third order, and higher, linear PDE, analogous to 3rd degree or higher algebraic equations and related geometric figures?

John
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  • The classification of second order PDE is based on the analytic geometry of conic sections. The equivalent third order PDE classification would be based the cubic equivalent of conic sections. This is studied under Algebraic Geometry and the shapes and properties are much less well-known. – John Oct 01 '18 at 04:46

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