探索非线性可积系统连续与离散系统的统一表达一直是可积系统领域研究难点之一。本项目通过引入Delta导数来建立时标非线性可积系统。一方面给出时标可积系统的Gel'fand-Dikii方法，建立时标可积系统的迹等式以及时标非线性可积系统的Hamilton结构等代数与几何性质。另一方面寻求时标系统的R-矩阵，系统地给出一条由Lax对、R-矩阵及“非线性化理论”去构造显式解的有效途径。同时，结合对时标经典方程的代数几何性质的研究，通过时标高维非线性耦合模型来分析多层流体中Rossby孤立波的解析解与孤立波的传播速度与频散关系，尤其是二维Rossby孤立波的耗散效应以及分析剪切流等对守恒律的影响，进而讨论海洋Rossby孤立波的演化规律，研究波-波，波-流等相互作用机理，解释海洋大气中的阻塞现象。该项研究不仅在理论上有助于解决可积系统判定问题，而且为认识海洋大气中复杂的非线性波动现象提供理论依据。

It is always a focus of integrable systems to explore the unified representation of continuous and discrete systems. In this project, the time scale nonlinear integrable system is established by introducing Delta derivative.On the one hand, we present the Gel’fand-Dikii method of time-scale integrable systems, and trace identity and the algebraic and geometrical properties of time-scale integrable systems,including the Hamiltonian structures,etc. On the other hand, we seek R- matrix of time scales and systematically give an effective way to construct soliton systems or explicit solutions of nonlinear evolution equations by Lax pairs, R-matrices and "nonlinear theory". At the same time, basing on the study of time-scale integrable systems and their properties, such as Boussinesq equation,Burgers equation, we analyze the analytical solution of Rossby solitary wave in multi-layered fluid and the relationship between the propagation velocity and dispersion of Rossby solitary wave, especially the dissipation effect of two-dimensional Rossby solitary wave and the influence of shear flow on conservation law by means of a time scale high-dimensional nonlinear coupled model.Then we discuss the evolution of Rossby solitary waves in the ocean atmosphere, the interaction mechanism of wave-wave and wave-current and the blocking phenomena in the ocean atmosphere. It is not only helps to solve the problem of integrable system determination in theory, but also provides theoretical basis for understanding the complex non-linear wave phenomena in the ocean and atmosphere.