Deriving the 2D KdV Equation for Overcoming Nonlinear Theory Challenges

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The discussion centers on the derivation of the 2D Korteweg-de Vries (KdV) equation, also referred to as the KP equation. The user has successfully completed the linear theory but is struggling with the weakly nonlinear theory, specifically in extending the derivation to include surface tension and an electrical field. They reference Johnson's 1980 paper on water waves and KdV equations as a valuable resource for overcoming their derivation challenges. Additionally, they note that while Johnson did not mention this paper in their prior communications, it has proven helpful in addressing their issues. The user has also found that the linear problem for the 3D case is comparable in difficulty to the 2D case, despite challenges in plotting solutions.
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Does anyone know of a derivation or has a reference to the derivation of the 2D KdV equation (known as the KP equation I believe). I have done the linear theory for this problem and the results look good but the next stage is the weakly nonlinear theory and I am having trouble with a certain aspect of it.
 
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Johnson 1980 Water waves and Kortweg de Vries equations. J Fluid Mech, 97, 701-19
 
Okay, I can work with this, Cheers. As an aside I am trying to extend the derivation to include the effect of surface tension and an electrical field. I have done this for one dimension but I have yet to do this for two.

The odd thing is that I was in contact with Johnson about this and he never mentioned this paper of his, weird.
 
I haven't seen the paper itself - it came from a footnote at the bottom of page16 "for a review of one and two dimensional KDV equations..." of Drazin and Johnson.
The book itself treats 2D but only in solutions not derivations.
 
It's actually quite a good paper, it tells me how I can go about overcoming my problem with the derivation and in that sense it's a very good thing. The linear problem for the 3D case actually wasn't much harder than the 2D case. What took me a while was plotting the solutions but I have not overcome that and I have some very pretty wave pictures.
 
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