Solution surfaces and surfaces from a variotional principle

In this thesis, we construct 2-surfaces in R 3 and in three dimensional Minkowski space (M3). First, we study the surfaces arising from modified Korteweg-de Vries (mKdV), Sine-Gordon (SG), and nonlinear Schr¨odinger (NLS) equations in R 3 . Second, we examine the surfaces arising from Korteweg-de Vries (KdV) and Harry Dym (HD) equations in M3. In both cases, there are some mKdV, NLS, KdV, and HD classes contain Willmore-like and algebraic Weingarten surfaces. We further show that some mKdV, NLS, KdV, and HD surfaces can be produced from a variational principle. We propose a method for determining the parametrization (position vectors) of the mKdV, KdV, and HD surfaces.