mKdV Surfaces

TL;DR

This paper explores the geometric properties of surfaces in three-dimensional space derived from solutions to the mKdV equation, providing explicit construction methods and linking them to variational principles involving curvature.

Contribution

It introduces a method to explicitly construct mKdV surfaces from solutions and connects these surfaces to variational principles involving curvature.

Findings

mKdV surfaces include Willmore-like and Weingarten surfaces

Explicit construction of surfaces from mKdV solutions

Some surfaces derive from a variational principle with curvature-based Lagrangian

Abstract

In this work, we consider 2-surfaces in ${\mathbb R}^3$ arising from the modified Korteweg de Vries (mKdV) equation. We give a method for constructing the position vector of the mKdV surface explicitly for a given solution of the mKdV equation. mKdV surfaces contain Willmore-like and Weingarten surfaces. We show that some mKdV surfaces can be obtained from a variational principle where the Lagrange function is a polynomial of the Gaussian and mean curvatures.