Korteweg–de Vries surfaces

We consider 2-surfaces arising from the Korteweg–de Vries (KdV) hierarchy and the KdV equation. The surfaces corresponding to the KdV equation are in a three-dimensional Minkowski (M3M3) space. They contain a family of quadratic Weingarten and Willmore-like surfaces. We show that some KdV surfaces can be obtained from a variational principle where the Lagrange function is a polynomial function of the Gaussian and mean curvatures. We also give a method for constructing the surfaces explicitly, i.e., finding their parameterizations or finding their position vectors.