一类微分差分可积方程的新精确解  

New explicit solutions to a differential-difference integrable equation

作　　者：陈守婷 高恒 李琪[2] CHEN Shouting;GAO Heng;LI Qi(School of Mathematics and Statistics,Xuzhou Institute of Technology,Xuzhou,Jiangsu 221018,China;Department of Mathematics,East China Institute of Technology,Fuzhou,Jiangxi 344000,China)

机构地区：[1]徐州工程学院数学与统计学院,江苏徐州221018 [2]东华理工大学数学系,江西抚州344000

出　　处：《石河子大学学报（自然科学版）》2020年第6期787-792,共6页Journal of Shihezi University(Natural Science)

基　　金：国家自然科学基金(11301454,11561002);江苏省六大人才高峰项目(2016-JY-081)。

摘　　要：作为一个著名的微分差分可积系统,Ablowitz-Ladik(AL)链由于其具有完全可积系统理论的支持,以及在非线性光学等领域中的应用,得到了广泛关注和研究,同时,微分差分可积方程的精确求解一直以来都是孤立子理论中的一个非常重要的课题,而朗斯基技巧是众多求解方法中一种高效直观的方法,因此,本文借助双Casoratian(离散形式的朗斯基)技巧和构造双Casorati行列式元素的矩阵方法,研究AL链一个具有双线性形式的微分差分方程,先将矩阵取成Jordan阵得到该方程具有双Casorati行列式形式的Matveev解,再将矩阵设成一个由特殊下三角矩阵和Jordan矩阵构成的准对角线矩阵形式,构造出具有双Casorati行列式形式的类有理解和Matveev解相互作用后的混合解,然后在将双Casorati行列式元素选取若干不同的形式后,得出Matveev解及其混合解在对应情况下的具体表达式。As a famous differential-difference integrable system,Ablowitz-Ladik(AL)lattice has been received considerable attention and investigation owing to its integrability and widely application in nonlinear optics and other fields.Searching for the exact solutions to the differential-difference integrable equations has always been a very important topic in the soliton theory.The Wronskian technique is an effective and direct approach for constructing the soliton solutions.In this paper,double Casoratian(discrete Wronskian form)technique and a matrix method for constructing the entries of the double Casorati determinant are applied to the differential-difference equation possessing the bilinear form related to AL spectral problem.The Matveev solutions in double Casoratian form are obtained by taking the matrix be a Jordan matrix.Furthermore,the interaction solutions between the rational-like cases and Matveev cases are constructed by taking the matrix as a quasi-diagonal form which consists of a special lower triangular matrix and a Jordan matrix.The obtained interaction solutions are also double Casorati determinant solutions to the differential-difference integrable equation.Moreover,the exact expressions of the Matveev solutions and interaction solutions are given under the conditions of different double Casorati determinant entries.

关 键 词：Ablowitz-Ladik链 微分差分可积方程 双Casoratian技巧 Matveev解 混合解 

分 类 号：O175.29[理学—数学]