Geometry, Partial Differential Equations And Their Symmetry Group

The motions of plane and space curve have a wide range of applications. Many interesting nonlinear evolution equations have been shown to be related to motions of curves. And many integrable equations arise naturally from motions of curve.We study partial differential equations (PDEs) by using symmetry group theory. The theory, also known as the classical Lie method of infinitesimal transformations, has been widely applied to the PDEs arising from mathematics and physics. From the finding of a symmetry group associated to the PDEs, we can obtain the corresponding optimal system and find the group-invariant solutions of the equations.In Chapter 1, we introduce the basic ideas of the symmetry group and we also recall the geometric curve flow and symmetry group history in briefly.In Chapter 2 is concerned the Lie point symmetry group of two type WKI equations based on the motions of plane and space curves by using the Lie classical method.The optimal systems of the obtained symmetry groups are discussed in chapter 3, and many interesting group-invariant solutions are also obtained.