半离散可积系统及其约化系统的研究

Compared to continuous integrable systems, the semi-discrete integrable systems are integrable differential-difference ones, for which the evolution equations are continuous in time, but are difference equations in a continuous spatial variable. Supported by the theory of completely integrable systems, various mathematical structures and applications in the fields of mathematical physics, quantum mechanics and biophysics, the semi-discrete systems have received considerable attention and investigation. Reduction systems are reduced by imposing the right linear combinations and reality conditions on the semi-discrete systems. In this project, we research on the semi-discrete systems and their reductions via Hirota method, double Casorati technique, inverse scattering problems, and so on. We construct many novel explicit solutions and integrable algebraic characters including symmetries, Lie algebraic structure, Hamilton struture and conservation laws. Furthermore, we establish the relationship between the semi-discrete and continuous cases. Our research will deepen the understanding to the semi-discrete integrable systems. We hope these novel cases will rich the soliton integrable algebraic characters.

相对于连续可积系统，半离散可积系统通常是指空间变量是离散的、而时间为连续的可积微分差分系统。鉴于其既具有完全可积系统理论的支持，又具有丰富的数学结构，而且在数学物理、量子力学、生物物理等众多领域都有应用，半离散系统一直受到广泛的关注与研究。通过对半离散系统选取适当的线性组合和约化条件，我们可得到它们的约化系统。本项目拟采用Hirota方法、双Casorati技巧、反散射变换等方法构造半离散可积系统及其约化系统的新的多种形式的精确解，研究其对称与Lie代数结构、Hamilton结构及守恒律等多个可积特征，并利用连续极限理论寻找它们与连续系统之间的联系。通过本项目的研究，将深化半离散可积系统的理解，希望这些新的模型会带来更加丰富的孤子可积代数特征。

本项目研究半离散可积系统及约化系统的求解和可积特征，取得了一系列的成果，列举如下：1）借助Hirota方法、双Casorati技巧、矩阵方法等分别对正向和负向的4个位势Ablowitz-Ladik等谱方程、等谱和非等谱的2个位势Ablowitz-Ladik方程等多个半离散可积方程进行构造与求精确解；2）利用广义双线性算子和Maple符号计算，构造出新的可积孤子方程，求出此类方程的lump解，并给出其动力学分析；3）通过双线性方法求解孤子方程位势约束后的有限维Hamilton系统，得到新的有限维Hamilton系统并给出求解该Hamilton系统的新思路；4）引进一个扰动孤子方程的新的Lax对，获得了具有递推关系的两组守恒律；5）发现了一个超可积的广义孤子方程族，给出其超双Hamilton结构，并提出了新的带自相容源的广义超可积方程族；6）构造新的约化系统并研究其可积性质。本项目的研究成果对可积系统中已有的经典方法产生了有益的补充与推广，并进一步丰富了孤子可积理论。