Algebraic Curves And Quasi-periodic Solutions To Soliton Equations

Based on the theory of algebraic curve, the thesis mainly studies quasi-periodicsolutions to soliton equations in terms of Riemann theta functional representa-tion. Four integrable systems considered here are Hu hierarchy, coupled modifiedKorteweg-de Vries hierarchy, Vakhnenko equation and a novel hierarchy of coupledequations. In order to extent the result to super integrable systems, the thesis alsotakes a super hierarchy of the vector Nonlinear Schro¨dinger equations and a superhierarchy of coupled derivative nonlinear Schro¨dinger equations into consideration.In chapter two, flows of the Hu hierarchy are straightened and quasi-periodicsolutions can be constructed. Utilizing the characteristic polynomial of Lax matrixin the stationary case, a two sheeted compact Riemann surface KNis introduced,on which the asymptotic expansions and divisor of the meromorphic function areobtained as well as the straightening formulas for flows of the whole hierarchy.Taking advantage of the representation for the the meromorphic function, Riemanntheta representations for the potentials are derived.In chapter three to five, quasi-periodic solutions for coupled modified Korteweg-de Vries hierarchy, Vakhnenko equation and a novel hierarchy of coupled equationscan be constructed. The three problems share one common feature in being asso-ciated with3×3matrix spectral problems which is diferent from2×2case formore complicated calculation and techniques. With the aid of the zero-curvatureequation, nonlinear evolution equations can be derived. Through the Lax matrix forstationary case, a three sheeted compact Riemann surface is introduced, on whichthe Baker-Akhiezer function and meromorphic functions are defined. On the ba-sis of divisors, asymptotic properties and Riemann-Roch theory, representations forBaker-Akhiezer function and meromorphic functions are obtained which can yield quasi-periodic solutions to the nonlinear evolution equations. Meanwhile, each chap-ter also has its own characteristics. In chapter three, the Riemann surface has threediferent infinite points which are not branch points. Only a little literature hasstudied three sheeted Riemann surface of this case. In chapter four, the Vakhnenkoequation is a reduction of a negative flow in the associated hierarchy. The intro-duced Riemann surface has one infinite point which is a threefold branch point. It isnecessary to consider the infinite point and the zero point simultaneously when an-alyzing the asymptotic extensions of meromorphic function. Chapter five is devotedto study a novel hierarchy of coupled equations. The associated Riemann surfacehas two infinite points, one of which is a twofold branch point, the other is not abranch point. Therefore, it is diferent from chapter three and chapter four both inchoosing local coordinate and calculating the arithmetic genus.In chapter six, two super hierarchies associated with diferent3×3spectralproblems are proposed and their super bi-Hamiltonian structures as well as infinitemany conservation laws are obtained.