Is there a way of generalising the KdV equation (1+1) to a 2D equation (2+1), with two spatial dimensions where the scales are the same for the spatial dimension? I'm aware of the Kadomtsev-Petviashvilli equation but the length scales are different for x and y. One of the things I like about the KP equation is that you can get a single equation for the free surface.

W. Choi has a formulation of the problem where he uses quantities on the surface and ends up with two equations for the velocity components on the surface. If the length scales in the x and y are the same, is it possible to obtain an equation for the free surface alone?