反谱变换方法在离散可积系统中的应用

The integrable discrete nonlinear equations have important applications in many fields such as nonlinear lattice dynamics, ladder type electric circuit and Volterra system. Recently, an integrated interdiscipline - progress in research on discrete integrable systems and differential geometry, statistics- has received extensive attention. Now an important task for the study of the discrete integrable systems is to find and develop some new tools. In this project, we will use the inverse spectral transform method to study the initial, boundary problems of the integrable discrete nonlinear equations, and to obtain the high-order soliton solutions, which are the deepening and expanding of my finished project supported by Youth Foundation of China. It is noted that the inverse spectral transform method used here mainly involves the dressing method based on Riemann-Hilbert problem and Dbar problem. We will confine ourselves to the discrete integrable equation such as discrete mKdV, discrete sine-Gordon and discrete three wave interaction equations. The research of the project will greatly enrich the theory of the integrable systems, and will contribute to the variety of the study and solutions of the initial and boundary problems of the discrete integrable equations. The selected topic in this project is one of the frontiers and core subjects in the research of the integrable systems, and has important value of science and extreme value of potential applications.

非线性离散可积方程在诸如非线性晶格动力系统、阶梯形电路以及Volterra系统等方面有着重要的应用。近年来，离散可积系统与微分几何学、统计学等方面的交叉引起人们的广泛关注。寻求和发展新的研究工具将是离散可积系统理论研究的重要任务。本申请项目将利用反谱变换方法对离散可积方程的初、边值问题以及高阶解等方面展开研究，其研究内容是对申请者所完成的青年基金项目的深化和拓展。这里的反谱变换方法主要包括Riemann-Hilbert问题和Dbar问题的穿衣服方法。本项目拟对离散mKdV，离散sine-Gordon，以及离散三波等方程开展相关研究。该项目的研究将极大地丰富可积系统的数学理论，并将促进离散可积方程初、边值问题的解法与解的形式多样性. 这些选题是当前可积系统研究的前沿和核心课题之一，有重要的科学价值和极大的应用价值。

反谱变换方法是研究非线性可积发展方程的重要方法之一。本项目利用反谱变换方法研究离散与连续非线性可积发展方程的相关问题。在离散问题方面分别研究了离散mKdV方程的高阶孤子解，Ragnisco-Tu方程的阶梯边值问题，二分量Ragnisco-Tu方程的反谱变换，离散六波方程的Dbar穿衣变换和一个广义Volterra格的反散射变换等问题。在连续可积系统方面，先后研究了带自相容源的三波方程，一类长短波方程，耦合Sasa–Satsuma系统，二维Boussinesq方程等。