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Coming from engineering rather than mathematics, I am recently dealing with non-linear partial differential equations e.g. like the well known Korteweg-de-Vries equation:

$$u_{t} + uu_x + u_{xxx} = 0$$

In the literature I found the two- and three-dimensional versions of the KdV-equation:

$$(u_{t} + uu_x + u_{xxx})_x + u_{yy} = 0 $$ $$(u_{t} + uu_x + u_{xxx})_x + u_{yy} + u_{zz} = 0$$

What bugs me now is how one derives the two- respectively three-dimensional versions from the original version? In all of the literature I have found so far, the multi-dimensional version seems to fall from heaven, nowhere I could find a complete derivation or any intuition behind it. This is unfortunately not the first time I can not grasp how one author jumps from the 1-D to a n-D version.

So generally asked: How does one derive a multi-dimensional version of a (non-linear)PDE given a 1 + time-dimensional form like the first equation? It would be more clear to me if the original version would be written in vector notation (with $\nabla$, $\cdot$, $\times$, $\Delta$, etc. operators). Would be great if someone with a more profound mathematical background could educate me.

Thanks a lot!

Edit:

The 2D version is also called Kadomtsev-Petviashvili equation, if that rings a bell for anyone.

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