半离散hungry型可积方程的非等谱推广及其分子解

Hungry-type integrable equations and their molecule solutions are currently a hot research topic because of their important application to the problem of computing eigenvalues of totally non-negative matrices. So far mainly fully-discrete equations have been considered in the literature and studies about the semi-discrete case and non-isospectral hungry-type equations are rare. Moreover, the existing molecule solutions of hungry-type equations, which are expressed in terms of a block Hankel determinant, are all restricted to zero boundary condition. This project will concentrate on semi-discrete hungry-type integrable equations. We will conduct an intensive study of non-isospectral extensions of these equations and corresponding molecule solutions via Hirota bilinear method and determinant techniques. Concretely, this project contains three parts. (i) Explore two semi-discrete hungry-type integrable equations whose molecule solutions will be expressed in terms of a block Toeplitz determinant. (ii) Extend several semi-discrete hungry-type integrable equations to non-isospectral case and investigate their molecule solutions with free boundary conditions, as well as their linearizations. (iii) Study a non-isospectral extension of a Bogoyavlensky lattice and design a related convergence acceleration algorithm. This project aims to expand the explorations of molecule solutions of semi-discrete hungry-type integrable equations and also to provide a new idea and theoretical basis for the studies on integrable numerical algorithms.

Hungry型可积方程及其分子解因在计算完全非负矩阵特征值方面有重要应用正成为时下研究热点。目前文献中主要考虑全离散方程情形，对半离散情形和非等谱hungry型可积方程的研究屈指可数，且已有hungry型方程分子解均是在方程边界为零的条件下得到，由块状Hankel行列式构成。本项目将以半离散hungry型可积方程为研究对象，采用Hirota双线性方法和行列式技巧，对这类方程非等谱推广及其分子解进行深入研究。具体内容包括：发展两个分子解由块状Toeplitz行列式构成的半离散hungry型可积方程；对几个半离散hungry型可积方程进行非等谱推广，并且探究它们在自由边界条件下的分子解和线性化表示；考察一个Bogoyavlensky格的非等谱推广，并设计与之相关的收敛加速算法。本项目旨在丰富半离散hungry型可积方程分子解的研究，并同时为可积数值算法方面的研究提供新思路和理论依据。

本项目采用Hirota双线性方法及行列式技巧在几个可积系统的分子解研究、非等谱推广、与正交多项式的联系及收敛加速算法方面都取得了一些创新性成果，部分结果已发表在国际知名期刊上。主要成果包括：获得了半离散hungry Lotka-Volterra方程及全离散Lotka-Volterra方程在自由边界条件下的分子解并阐明了它们与正交多项式间的联系；推广了Schur流、相对论Toda方程及Camassa-Holm方程族的第二流到非等谱情形并给出了它们的分子解、N-peakon解以及讨论了它们与正交多项式间的联系；得到了一个推广的ε算法用以递推计算推广的Shanks变换并找到了其与推广的全离散Lotka-Volterra方程之间的Miura变换。本项目的实施进一步丰富了可积系统的分子解研究，促进了可积系统与正交多项式及收敛加速算法方面的交叉研究，具有重要理论意义。