可积系统的双线性约化方法  

Bilinearization-reduction approach to integrable systems

作　　者：张大军[1] Zhang Da-Jun(Department of Mathematics,Shanghai University,Shanghai 200444,China)

机构地区：[1]上海大学数学系,上海200444

出　　处：《物理学报》2023年第10期20-31,共12页Acta Physica Sinica

基　　金：国家自然科学基金(批准号:12271334,12126352,11875040)资助的课题.

摘　　要：本综述主要介绍了双线性约化方法在可积系统求解中的应用.这一方法基于双线性方法和解的双Wronskian表示.对于通过耦合系统约化而获得的可积方程,先求解未约化的耦合系统,给出用双Wronskian表示的解;进而利用双Wronskian的规则结构,施以适当的约化技巧,获得约化后的可积方程的解.以非线性Schrödinger方程族和微分-差分非线性Schrödinger方程为具体例证,详述此方法的应用技巧.除了经典可积方程,该方法也适用于非局部可积系统的求解.其他例子还包括Fokas-Lenells方程和非零背景的非线性Schrödinger方程等可积系统的求解.The paper is a review of the bilinearization-reduction method which provides an approach to obtain solutions to integrable systems.Many integrable coupled systems can be bilinearized and their solutions are presented in terms of double Wronskians(or double Casoratians in discrete case).The bilinearization-reduction method is based on bilinear equations and solutions in double Wronskian/Casoratian form.For those integrable equations that are reduced from coupled systems,one can first solve the unreduced coupled system,obtaining their solutions in double Wronskian/Casoratian form,then,implement suitable reduction techniques,so that solutions of the reduced equation can be obtained as reductions of those of the unreduced coupled system.The method proves effective in solving not only classical integrable equations but also the nonlocal ones.The so-called nonlocal integrable equations were introduced by Ablowitz and Musslimani via reductions with reverse-space(or reverse-time,or reverse-space-time).Note that this method particularly provides a convenient bilinear approach to solve nonlocal integrable systems.In this review,the nonlinear Schrödinger hierarchy and the differential-difference nonlinear Schrödinger equation are employed as demonstrative examples to elaborate this method.These two examples will be pedagogically helpful in understanding the reduction technique.The reduction is implemented by imposing suitable constraints on the basic column vectors of the double Wronskian/Casoratian.Realizations of the constraints are converted to solve a set of matrix equations which varies with the constraints.Special solutions of the matrix equations are provided,which are also helpful in understanding the eigenvalue structure of the involved spectral problems corresponding to the considered equations.Other examples include the Fokas-Lenells equation and the nonlinear Schrödinger equation with nontrivial background.Since many nonlinear equations with physical significance are integrable as reductions of integrable cou

关 键 词：双线性约化方法 双 Wronski 行列式 可积系统 精确解 

分 类 号：O175.29[理学—数学]