On The Structure Equations Of Lie Pseudo-Groups And Differential Invariant Of Some Evolution Equations

The theory of Lie group is an important mathematical field in the 20th Century, By com-bining the powerful theory of Lie group, Olver and Pohjanpelto have successfully extended the theory of equivariant moving frames. By using the theory of equivariant moving frames, on one hand, they have developed a practical algorithm for determining the Maurer-Cartan structure equations of Lie pseudo-groups. on the other hand, the theory of equivariant moving frames is a powerful tool for determining a generating set of the differential invariant algebra of Lie pseudo-groups. As we all know, nonlinear evolution equation is an important field in the contem-pary study of nonlinear physics, especially in the study of soliton theory. The research about nonlinear evolution equation is helpful in clarifying the movement of matter under the nonlin-ear interactivities and plays an important role in scientifically explaining of the corresponding physical phenomenon and engineering application.In this paper, with the aid of symbolic computation and the theory of equivariant moving frames, we mainly study the Maurer-Cartan structure equations and Cartan structure equations of Lie pseudo-groups; the Maurer-Cartan structure equations of nonlinear evolution equation and generating set of the differential invariant algebra. The article consists of the following parts:Chapter 1 introduces the backgound, development and research methods of the theory of Lie group, structure theory of Lie pseudo-groups, the theory of equivariant moving frames and the classical invariant theory.Chapter 2 states the babsic thought and applications of the theory of "AC=BD ". Using this theory we describe other common research methods.In Chapter 3, structure theory of Lie pseudo-groups are illustrated. One is Cartan structure theory and the other is Maurer-Cartan structure theory. At last, we derive that the two structure theories are isomorphic, illustrated by an example.Chapter 4 studies the theory of equivariant moving frames, we derive a generating set of the differential invariant algebra of Lie pseudo-groups of nonlinear partial equation, and extend the theory of equivariant moving frames to infinite-dimensional Lie pseudo-groups. Finally we describe the algorithm for specific examples.In chapter 5, first, we search the relationship between Maurer-Cartan structure equations of Lie pseudo-groups and structure equations of Lie algebra; second, we further discuss the relationship between the differential invariant and Maurer-Cartan structure equations, along with the connection between Backlund transformation and Maurer-Cartan structure equations of nonlinear partial equation. At last a short summary of the dissertation is given.