Quasi-periodic Solutions Of The Soliton Hierarchies Associated With3×3Matrix Spectral Problems

The thesis can be mainly divided into two parts. First, with the help of Lenard recursion equations and the zero-curvature equation, we derive a new super KdV hierarchy and super KN hierarchy related to two even3×3matrix spectral problems. Moreover, generalized bi-Hamiltonian structures and infinite conservation laws of these two hierarchies are established; On the other hand, based on the theory of trigonal curve and the knowledge of algebraic geometry, we construct the quasi-periodic solutions of three hierarchies of soliton equations associated with three3×3matrix spectral problems.In chapter two, by extending the corresponding classical2×2spectral problems to the even3×3super matrix spectral problems according to super Lie algebra, we propose a new super KdV hierarchy and super KN hierarchy resorting to the compatible condition i.e. zero-curvature equation. Applying the super trace iden-tity, we discuss the generalized bi-Hamlitonian structures of the two super soliton hierarchies. Furthermore, we establish the infinite sequence of conserved quantities of the super KdV equation and super KN equation.As we all know, quasi-periodic solutions of soliton equations not only reveal in-herent structure mechanism of solutions and describe the quasi-periodic behavior of nonlinear phenomenon or characteristic for the integrability of soliton equations, but also can be reduced to find multi-soliton solutions, elliptic function solutions, and others. Therefore, the research on the quasi-periodic solutions of soliton equations is of greatest importance. From chapter three to five, we discuss the quasi-periodic solutions of three hierarchies of soliton equations associated with three different3×3matrix spectral problems, respectively. With the aid of the characteristic polynomial of Lax matrix for soliton hiearachy, we define a trigonal curve κg, and then its com- pactification becomes a three-sheeted Riemann surface of arithmetic genus g. We introduce the appropriate Baker-Akhiezer function, meromorphic function and ellip-tic variables on the three-sheeted Riemann surface, from which soliton equations are decomposed into the system of solvable Dubrovin-type ordinary differential equa-tions. Then, under the Abel map, the flows of soliton hierarchy are straightened. Furthermore, in accordance with the properties of the zeros and singularities of the meromorphic function and Baker-Akhiezer function, we get their Riemann theta function representations by means of the second and third Abel differentials, Rie-mann theorem and Riemann-Roch theorem. Combing the Riemann theta function representations of the meromorphic function or(and) the Baker-Akhiezer function with their asymptotic properties, we finally obtain the quasi-periodic solutions of soliton equations.