Quasi-periodic Solutions To Some Continuous And Discrete Soliton Hierarchies

The thesis mainly aims to study the application of trigonal curve theory to the integrable systems.We discuss the basic properties for the derived trigonal curve and obtain quasi-periodic solutions for the continuous and discrete soliton equations associated with 3×3 matrix spectral problem.The hierarchies include the coupled Burgers hierarchy,a new hierarchy of coupled equations,the mod-ified Blaszak–Marciniak lattice hierarchy and the coupled Bogoyavlensky lattice hierarchy.Starting with a 3×3 matrix spectral problem,a hierarchy of soliton equations is given by means of the zero-curvature equation and Lenard recursion relations.With the help of the characteristic polynomial of Lax matrix,a trigonal curve is introduced.Adding the appropriate infinite point,the compactified curve₂)becomes a three-sheeted Riemann surface,on which the Baker–Akhiezer function and meromorphic functions are defined.And we study the construction of three kinds of Abelian differentials.We analyze the asymptotic properties of the mero-morphic functions and the Baker–Akhiezer function.The divisors of the meromor-phic functions are defined by introducing the elliptic variables.Meanwhile,the zeros and poles of the Baker–Akhiezer function are proposed.The solutions of the Baker–Akhiezer function and the meromorphic functions are constructed in terms of Riemann theta function by virtue of Abelian differentials and Riemann–Roch theorem.Comparing theta function expressions for the Baker–Akhiezer function and the meromorphic functions with their asymptotic expansions,we can derive the quasi-periodic solutions to the continuous or discrete soliton hierarchy.In chapter two and three,we discuss the coupled Burgers hierarchy and a new hierarchy of coupled equations related to 3×3 continuous matrix spectral problems.In the last two chapters,by considering 3×3 discrete matrix spectral problems related to negative flows,we derive the modified Blaszak–Marciniak lattice hierarchy and the coupled Bogoyavlensky lattice hierarchy.Though the associated three-sheeted Riemann surfaces to four 3×3 matrix spectral problems all have two infinite points,the properties of the zero points are different in the discrete case.The infinite points and zero points should be considered when we study the asymptotic expressions of the meromorphic functions and the Baker–Akhiezer function for the discrete spectral problem.In chapter four,the Riemann surface has three zero points,while the associated algebraic curve in chapter five has two zero points.So we shall choose different local coordinates to analyze the asymptotic expansions of the Baker–Akhiezer function and the meromorphic functions.