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

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Discussion Overview

The discussion revolves around the classification of higher-order partial differential equations (PDEs) that involve multiple independent variables. Participants explore whether there exists a systematic classification similar to that of second-order PDEs, such as hyperbolic, parabolic, and elliptic classifications, and the complexities involved in creating such a system.

Discussion Character

  • Exploratory, Debate/contested, Technical explanation

Main Points Raised

  • One participant questions if there is a classification system for PDEs with an arbitrary number of independent variables and of order n.
  • Another participant suggests that the Monge cone is relevant for classifying second-order PDEs in multiple variables, but notes uncertainty regarding higher-order classifications.
  • Some participants express surprise at the existence of equations that may not fit into any classification system, raising questions about the complexity and purpose of such classifications.
  • There is a distinction made between classifying second-order PDEs in n variables versus classifying PDEs of order n with varying numbers of independent variables.
  • Participants acknowledge uncertainty regarding the classification of higher-order PDEs with multiple independent variables.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the existence of a comprehensive classification system for higher-order PDEs. Multiple competing views are presented regarding the applicability of the Monge cone and the challenges in classifying equations of different orders and variable counts.

Contextual Notes

The discussion highlights limitations in understanding the classification of PDEs, particularly regarding the assumptions involved and the potential complexity of creating a unified classification system.

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.
 

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