多维相容离散系统的达布变换和可积性研究

In the development of the discrete integrable systems, there were lots of breakthroughs in the past 20 years. The discrete integrable system is one of the hotspots and focus in the research area of integrable systems. It plays a key role in the development of other research areas, such as discrete geometry, discrete complex analysis, and so on. As an understanding of the integrability, the property of multidimensional consistency provides a new angle for the research of discrete integrable systems. The project here will do some further study on the discrete systems with multidimensional consistency, including three aspects: Firstly, study the basic Darboux and binary Darboux transformation of the Lax pair obtaining from the multidimensional consistency, discuss their N-step forms, and then give solutions in determinant form, such as the N-soliton solution. Then study the inverse scattering method of the non-autonomous multidimensionally consistent discrete integrable systems. At last, use the conserved quantity coming from the inverse scattering method to study the integrable characteristics, such as the conservativeness, of the multidimensionally consistent systems with non-zero asymptotic behavior of its potential. This project not only supply new ways and theory of solving discrete integrable systems, but also uncover the theoretical significance of the inverse scattering method and Darboux transformation, which is more important, for further understanding of the integrability of multidimensionally consistent system. This project is also important for the development of discrete integrable systems and its related theory.

离散可积系统的发展在过去20多年来产生了很多突破性成果，是目前可积系统领域研究的热点和重点之一，为离散几何、离散复分析等领域的发展起到关键推动作用。多维相容性作为可积性的一种理解为离散可积系统的研究提供新的视角。本项目将对具有多维相容性的离散系统作进一步研究，具体包括三个方面：研究由多维相容性导出的Lax对所对应的初等达布变换和二元达布变换, 探讨其多步形式，给出多孤子解等精确解的行列式表示；研究非自治多维相容离散可积系统的反散射方法；利用反散射变换过程中提供的守恒量等可积特征，研究位势具有非零渐近性的多维相容系统的守恒性等可积特征。本项目不仅是对目前离散可积系统求解途径及理论的发展，更重要的在于揭示反散射方法和达布变换的理论意义，对进一步认识多维相容系统的可积性，发展离散可积系统相关理论具有重要意义.

离散可积系统是指时间、空间变量均为离散变量的可积差分方程，是目前可积系统领域研究的热点和重点之一。本项目对具有多维相容性的离散可积系统作研究，研究其初等达布变换和二元达布变换并探讨其多步形式，给出精确解的行列式表示；研究非自治多维相容离散可积系统的反散射方法；研究多维相容系统的可积特征；非局部方程的精确求解方法。重要研究成果有：基于Hirota-Miwa方程和周期约化条件，构造双线性、非线性、线性三种表示形式的链势修正KdV方程和二分量链势修正Boussinesq方程，同时利用达布变换方法，得到其精确解的行列式表示。该结果可推广到N分量的链势Gel’fand-Dikii方程族。基于谱问题和散射数据，研究构造多维相容系统无穷守恒律的方法。基于双线性方法和非局部约化条件，构造多种不同形式的非局部带导数非线性薛定谔方程及其精确解。基于双线性方法和KP方程族的约化，得到非局部Schrodinger-Boussinesq方程的孤子解。利用约化技巧，从链KP方程的有理解我们推导出了链势KdV方程和两个半离散链势KdV方程的有理解。这些研究使我们进一步认识了多维相容系统的可积性。