Integrable Semi-discretisation Of The Drinfel’d-Sokolov Hierarchies

This thesis is concerned with algebraic construction of the semi-discrete Drinfel’d-Sokolov hierarchies associated with affine Lie algebras A_r^（1）,A_2r^（2）,C_r^（1）and D_r+1^（2）and their integrability characteristics.Chapter 1 is the introduction of the thesis,including the research background,the current progress of the research of integrable discretisation as well as the main results of the thesis.The general framework of the direct linearisation approach is given in Chapter 2,which mainly covers the notion of infinite matrix and also the infinite matrix represen-tation of the direct linearisation.In Chapter 3,by introducing a plane wave factor describing discrete odd-flows,dynamics of the infinite matrix are established,and subsequently,a novel semi-discrete Kadomtsev–Petviashvili equation is constructed within the infinite matrix framework,together with its representations in terms of theτ-function and Lax matrices.Symmetry and periodic reductions of the semi-discrete Kadomtsev–Petviashvili equation are discussed in Chapter 4,from which new integrable semi-discrete equations of B_∞-,C_∞-,A_r^（1）-,A_2r^（2）-,C_r^（1）-and D_r+1^（2）-types are obtained.Moreover,the structure of Z_N-graded matrices with fractional form is observed in Lax pairs of these equations.Chapter 5 is concerned with continuum limits of the semi-discrete Kadomtsev–Petviashvili equation and the semi-discrete Drinfel’d-Sokolov hierarchies.Two differ-ent limit schemes towards the Korteweg–de Vries-type and two-dimensional Toda-type systems are established.In Chapter 6,by selecting special integration measures in the linear integral equa-tion,the Cauchy matrix formulae of finite-pole solutions for these semi-discrete equa-tions are given,which further guarantees their integrability.Concluding remarks and plans for further research are given in Chapter 7.