Derivation of 2D Korteweg-de-Vries equation

Question

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.

Answer 1

I'm not sure how one would derive the analog, but my gut feeling is that one would generalize the KdV hamiltonian (which is the second (aka "energy") integral of motion on the wikipedia page). I'm not sure if Hamiltonian mechanics is something you've covered in engineering courses, as it can quickly turn into very advanced/technical pure mathematics, but the legend Jerrold Marsden was an absolute pioneer of not only developing this advanced pure math, but also applying it to real engineering problems (he was a Professor of Engineering and Control & Dynamical Systems at CalTech). There is a great lecture of his on YouTube that highlights applications of his research to problems in engineering.

There is a "2D analog" of KdV, it's called the "Kadomtsev–Petviashvili Equation" (KP). Like KdV, it is integrable (it has countably infinitely many conserved quantities/integrals of motion). I wonder if you took the Hamiltonian for the KP equation and only kept one spatial dimension, if it would be equal to the KdV Hamiltonian. Perhaps I shall try it!

Answer 2

The KP equation was derived from physical principles, so you should be able to find this derivation somewhere. Worst case scenario, you may have to refer to the original paper of Kadomtsev and Petviashvili, who derived it for plasma waves; however, I believe that paper is in Russian. And it's probably not helpful if you aren't familiar with plasma physics. Later, Ablowitz and Segur derived it for surface and internal water waves, and Pelinovsky, Stepanyants and Kivshar derived it in nonlinear optics, so you could look for references in those directions. It is better to think of the KP equation as physical equation in its own right, rather than just as a purely mathematical generalization of the KdV equation. After all, KdV can be generalized in any number of ways depending on which of its properties you intend to preserve.

Typically one does not just generalize a PDE to higher dimensions without first thinking about what feature of the PDE you are trying to generalize. For example, there is another generalization of KdV to two spatial dimensions called the Novikov-Veselov equation, which is related to but not the same as the KP equation. I understand the Novikov-Veselov equation to have originally arisen more out of some mathematical investigations on inverse scattering theory rather than from physical principles. Thus I would also expect the 3D case to arise either from physical applications or from attempting to generalize some aspect of the inverse scattering theory for the KdV or KP equations, as this is likely the area of research on these equations that has received the most attention. But I cannot say for sure, not being an expert on integrable systems.

See here for a brief history of the KP equation, where I found the references for the physical derivations.