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Consider the second-order nonlinear PDE

$$(\partial_x\partial_y\varphi)\cdot\varphi = \partial_x\varphi\,\partial_y\varphi.$$

This PDE is solved by all ('separable') functions $\varphi\in C^2(\Omega)$ of the form $\varphi(x,y) = \ell(x)\cdot r(y)$ (any $\Omega\subseteq\mathbb{R}^2$ open).

I'd like to know whether each (classical) solution of the PDE is of this form.

Are you aware of any 'uniqueness' results for the above PDE which might (dis)prove this?

Improve this question
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    $\begingroup$ This equation is equivalent to $\partial_x \partial_y \log(\varphi) = 0$. This is the 2-dimensional wave equation for $\log(\varphi)$ in "null" coordinates. $\endgroup$
    – Igor Khavkine
    Commented Jul 15, 2020 at 9:57
  • $\begingroup$ That settles it, thanks. @IgorKhavkine $\endgroup$
    – fsp-b
    Commented Jul 15, 2020 at 10:19
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    $\begingroup$ @fsp-b Not completely: it shows that it is true for positive solutions, but since the equation has compactly supported solutions, you can combine them to form solutions that aren't (globally) separable. $\endgroup$
    – Martin Hairer
    Commented Jul 15, 2020 at 10:27
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    $\begingroup$ Many thanks for your comment, @MartinHairer I in fact happen to be interested in positive solutions only, so for me, this in fact gives what I needed. $\endgroup$
    – fsp-b
    Commented Jul 15, 2020 at 10:35

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