New Integrable Models With N-Peakons And Algebro-Geometric Solutions Of Soliton Equations

The thesis can be mainly divided into two parts. First, the derivation of hier-archies of new nonlinear evolution equations and their infinitely many conservation laws. On the other hand, the construction of algebro-geometric solutions of soliton equations.Soliton equations are nonlinear partial differential equations described by infinite-dimensional integrable systems and have various beautiful algebraic and geometric properties. An interesting problem of common concern is to search for new soliton hierarchies associated with spectral problems, which is usually very difficult. There are at least two important applications:(i) it is used to obtain a considerable number of new integrable models; (ii) these soliton equations have potential applications. In addition, a basic property of soliton equations is that they possess infinitely many conservation laws. On the other hand, the existence of infinitely many conserva-tion laws for soliton equations further confirms their integrability. In chapter two to four, three hierarchies of new nonlinear evolution equations associated with different spectral problems and their infinite sequences of conserved quantities are obtained, respectively. Negative flows in the hierarchies derived in chapter two and three ad-mit exact solutions with n-peakons, their dynamical systems can be constructed with the help of the distribution approach.The study of explicit solutions for soliton equations has been very important in the areas of mathematics and physics. In the fifth chapter, N-soliton and algebro-geometric solutions of the well-known KdV6 equations are constructed. From the beginning of this chapter, a hierarchy of new nonlinear evolution equations which contains the KdV6 equation is proposed. With the aid of scattering coordinates and the inverse scattering method, N-soliton solutions of the first three equations (including the KdV6 equation) in this hierarchy are derived. Based on the Lax representations of stationary evolution equations, the first three equations are re-spectively reduced to solving two solvable ordinary differential equations. The Abel-Jacobi coordinates are introduced to straighten the corresponding flows, from which algebro-geometric solutions of the first three equations are constructed in terms of Riemann theta functions.In chapter six, resorting to the inverse scattering method and algebro-geometric approach, N-soliton and algebro-geometric solutions of two mKdV type equations are obtained. To avoid the limit of using the Riemann theorem, after straightening the corresponding flows under the Abel-Jacobi coordinates, algebro-geometric solu-tions of the two mKdV type equations are constructed according to the asymptotic properties and the algebro-geometric characters of the meromorphic function (?) and hyperelliptic curveΚn.Using the same algebro-geometric approach as in chapter six, algebro-geometric solutions of three mixed AKNS equations are derived in the light of the asymptotic properties and the algebro-geometric characters of the meromorphic function (?), Baker-Akhiezer vectorψand hyperelliptic curveΚn in the last chapter.