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In Section 4.4 of Partial Differential Equations by Evans, the author describes several techniques for converting certain nonlinear equations into linear equations. First, the author introduces the Cole-Hopf transformation that maps the Burgers' Equation into the Heat Equation. This enables solving the Burgers' Equation indirectly by solving the corresponding Heat Equation and applying the inverse Cole-Hopf transformation. He also describes potential functions and Hodograph and Legendre transforms for other nonlinear PDEs.

My main question is the following: Which nonlinear PDEs can (or can not) be converted into linear PDEs? Is there a particular class of nonlinear PDEs for which a nonlinear transformation that transforms the equation into a linear PDE exists? I am not looking for a general method of transforming nonlinear PDEs, but if such nonlinear transformations exist. I would also be grateful for any relevant literature.

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  • $\begingroup$ I suspect the answer to the fully-general question is "all of them, but not necessarily with respect to a useful coordinate system." Answering in more detail means figuring out how to formalize one's notion of a coordinate system being useful/well-behaved... $\endgroup$
    – user3716267
    Commented Jun 25 at 14:21
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    $\begingroup$ I think Evans or Strauss said that there are Infinite types of PDE Cases & Classification is "Impossible". With that , I suspect that nobody "can know" which Classes are convertible & which are not convertible because the Classes are unknown. We may talk about "Individual Known Cases" to show the conversions or to show that those can not be converted. $\endgroup$
    – Prem
    Commented Jun 25 at 15:09
  • $\begingroup$ @Prem Yes. I am also interested to know if any PDEs cannot be converted. $\endgroup$
    – user572780
    Commented Jun 25 at 15:12
  • $\begingroup$ If you don't get a satisfactory answer here, I suggest you post this on MathOverflow. $\endgroup$
    – Deane
    Commented Jun 25 at 15:37
  • $\begingroup$ @Prem, it's well known that most PDEs are not any of the standard types (elliptic, hyperbolic, parabolic) and that there is no possible way to classify, say, linear PDEs by their properties. But this is a separate issue from the question of when a nonlinear PDE can be transformed into a linear one. Here, there is a possibility that there is a systematic way of determining this. If so, there are not many people who would know about this. That's why I suggested migrating this question to MathOverflow. $\endgroup$
    – Deane
    Commented Jun 26 at 2:50

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