离散可积系统

研究与4×4离散矩阵谱问题相联系的Lax矩阵导出的四方曲线及紧化给出的四叶Riemann面，讨论Baker-Akhiezer函数和亚纯函数的性质。引入Abel坐标并拉直离散流和连续流。发展一个系统的方法求解与4×4离散矩阵谱问题相联系离散孤子方程的代数几何解。寻找新的可积尖孤子方程和相应有限维可积系统，进而揭示隐藏的新正交性。研究2+1维可积系统的可积离散化，设计相应的数值格式进行数值模拟。围绕寻找新的序列变换及相应的可积加速收敛算法开展研究。研究连分式理论与离散可积系统的联系和通过离散可积系统计算pfaffian的数值算法等新问题。探讨Hamiltonian、Hirota离散化、直接线性化等多种经典可积离散化方法的推广，研究它们在超可积/超对称可积系统研究中应用。研究超对称格点规范理论、超对称正交多项式与超对称离散可积系统的关系。构造新超离散可积和超对称离散可积系统并研究它们的性质。

The theory of solitons and integrable systems allows one to study a wide range of phenomena and problems arising from physics, biology, and pure and applied mathematics. The development of soliton theory and integrable systems has profoundly affected many scientific disciplines, e.g., physics, mathematics, computer science, biology, chemistry, and etc., offering a new understanding of many nonlinear phenomena. Besides, integrable systems have been found to be associated with many areas in mathematical physics. Integrable systems also bridge strong interdisciplinary links among these areas...The theory of tetragonal curves and its application in discrete integrable system are studied, from which a systematic method is developed to construct the algebro-geometric solutions of discrete soliton equations associated with 4×4 discrete matrix spectral problems. The Lax matrices related to the 4×4 discrete matrix spectral problems lead to tetragonal curves and four-sheeted Riemann surfaces by their compactification. Then we introduce the meromorphic functions and Baker–Akhiezer functions on the four-sheeted Riemann surfaces and discuss their properties. A basis for holomorphic differentials are constructed on the four-sheeted Riemann surfaces. The asymptotic expansions for three kinds of Abel differentials are given near infinite points and zero ones. The Abel map, Abel-Jacobi coordinate and meromorphic differential on the four-sheeted Riemann surfaces are introduced to straighten discrete flow and continuous flow. On basis of the expansions of the Riemann theta function representation for meromorphic functions and Baker–Akhiezer functions, we obtain algebro-geometric quasi-periodic solution solutions of discrete soliton equations with physical background associated with 4×4 discrete matrix spectral problems. New integrable peakon equations and the corresponding finite dimensional integrable systems will be found to reveal the hidden new orthogonality. We study the integrable discretization of some 2+1-dimensional integrable systems with physical background and designs corresponding numerical schemes for numerical simulation. The focus is to find a new sequence transformation and the corresponding integrable accelerated convergence algorithm. Besides, some new issues such as link between continuous fraction theory and discrete integrable systems, and the numerical algorithm of computing Pfaffian via discrete integrable systems etc. will be touched upon. Typical methods of discretization such as the Hamiltonian method, Hirota method, direct linearization method, Backlund/Darboux transformation approach will be examined and generalized so that they may be applied to the study of super and supersymmetric integrable systems. The interplay between the lattice gauge theory and the supersymmetric orthogonal polynomials and supersymmetric integrable systems will be persuaded. In this way, new discrete integrable systems of super andsupersymmrtric types will be obtained and thier properties will be studied.