Applications Of Trigonal Curve To The Discrete Integrable Systems

Based on the theory of trigonal curve,the thesis concerns finite genus solutions of soliton hierarchies associated with 3 × 3 discrete matrix spectral problems.Four discrete integrable systems taken into account in this paper are Itoh–Narita–Bogoyavlensky lattice hierarchy,modified Belov–Chaltikian lattice hierarchy,four-component Toda lattice hierarchy and Merola–Ragnisco–Tu lattice hierarchy.Starting with a 3 × 3 discrete matrix spectral problem,soliton hierarchy is derived with the help of the discrete zero-curvature equation and Lenard recursion equations.Resorting to the characteristic polynomial of Lax matrix for the stationary case,we introduce a trigonal curve,on which the Baker–Akhiezer function and associated meromorphic functions are defined.The compactified curve 2)becomes a three-sheeted Riemann surface of arithmetic genus 2).We introduce the elliptic variables by defining the zeros and poles of the meromorphic functions,then the discrete soliton hierarchy is decomposed into the system of solvable Dubrovin-type ordinary differential equations.Furthermore,asymptotic properties and divisors of the meromorphic functions and Baker–Akhiezer function are studied.Then the Riemann theta function representations of the meromorphic functions and Baker–Akhiezer function are constructed by means of three kinds of Abelian differentials and Riemann–Roch theorem,from which we can obtain finite genus solutions of the discrete soliton hierarchy.The four lattice hierarchies share one common feature,which are all associated with 3 × 3 discrete matrix spectral problems and three-sheeted Riemann surfaces.It is necessary to consider the infinite points and the zero points simultaneously when analyzing the asymptotic expansions of meromorphic functions and Baker–Akhiezer function,which is different from 3 × 3 continuous case.Meanwhile,each chapter has its own characteristic.In chapter two,the Riemann surface has two different infinite points(one is double branch point and the other one is not branch point)and a threefold zero point.In chapter three,the Riemann surface has three different infinite points(they are not branch points)and a threefold zero point.In chapter four,the Riemann surface has three different infinite points and it is not necessary to consider the zero points for absence of the meromorphic function and Baker–Akhiezer function's singularities.In chapter five,the associated Riemann surface has three infinite points and three zero points.It is different in calculating the arithmetic genus and choosing local coordinates.So are the asymptotic properties of the meromorphic functions and the Baker–Akhiezer function.