Curve Flows, Pseudo-spherical Surface And Multi-component Integrable Systems

The present thesis is devoted to study the relationship between the Multi-component integrable systems and the geometric structure of the Pseudo-spherical surface and the curve flows in several homogeneous space such as centro-symplectic space, two-dimensional conformal sphere, the n-dimensional unite sphere etc. These Multi-component integrable systems includes the well-known matrix KdV equation,two-component and three-component Camassa-Holm-Hunter-Saxton systems and some new peakon type multi-component in-tegrable systems. We study their geometric realization,geometric construc-tion,geometric integrability and some associated problems such as:bi-Hamilton structure,Lax pair,conservation laws,nonlocal symmetries and peaked solu-tions etc.The main works of this dissertation are in three parts as follows.1. We study the curve theory in the centro-symplectic geometry from the point of view of Klein geometry. Based on the symplectic-orthogonalization pro-cess and the Fels-Olver's equivariant moving frame method, we obtained the sym-plectic arc-length,the symplectic curvatures,the Serret-Frenet formulas and the fundamental theorem for the regular star-shaped curves in the symplectic space; We compute the complete set of generating differential invariants for the curves in the symplectic space under the linear action of the symplectic group; The nat-ural frame is also obtained by a gauge transformation. Using the natural frame, We study the curve motions in the four-dimensional centro-symplectic geome-try. We show that the 2×2 matrix KdV equation and a Jordan KdV system as long as their hamiltonian structure naturally come from the curve motions in the four-dimensional centro-symplectic geometry. These results provide a nice in-terpretation for the geometric origin of the matrix KdV equation and the Jordan KdV system, extend the results of professor Chou Kai-Seng and Qu Chang-Zheng that KdV equation naturally come from the curve motion in centro-affine plane.2. We study the nonlocal curve motion in the two kind of homogenous space: the two-dimensional conformal sphere and the the n-dimensional unite sphere. From the analysis of the Cartan structure equation of the curve flow, we con-structed several new multi-component peakon systems. Their integrability are also investigated. We show that the complex Camassa-Holm system and the com-plex Hunter-Saxton systems naturally come from the motion of parametric curve on the two-dimensional conformal sphere. They are new two-component extension of the classical Camassa-Holm equation and the complex Hunter-Saxton equation. It is shown that both of them admit bi-Hamiltonian structure and Lax pair of con-formal type. Hence they are completely integrable. Another new integrable model we derived come from the curve motion on the n-dimensional unite sphere. It is the vector Olver-Rosenau-Qiao eqaution. This equation is an interesting vector version extension of the classical Olver-Rosenau-Qiao equation. We show that it is also integrable in the sense of Lax pair.3. We study the integrability of some well-known multi-component peakon systems in geometric way. It is shown that two-component and three-component Camassa-Holm-Hunter-Saxton systems are geometrically integrable, namely they describe one parameter of Pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws are directly computed. Their nonlocal sym- metries depending on the pseudo-potentials are also obtained. It is shown that the systems admit infinite number of nonlocal symmetries. We also proposed a new two-component Olver-Rosenau-Qiao eqaution. The new system is proven geomet-rically integrable and possess infinite number of conservation laws. Furthermore, exact solutions such as peakon solutions are also obtained.