可积系统的双线性约化方法

张大军

doi:10.7498/aps.72.20230063

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.

关键词:

Abstract

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 coupled systems, the paper provides a review as well as an introduction about the bilinearization-reduction method that can be used to solve these nonlinear integrable models.

Keywords:

作者及机构信息

张大军

上海大学数学系, 上海　200444

通信作者: 张大军, djzhang@staff.shu.edu.cn

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

Authors and contacts

Zhang Da-Jun

Department of Mathematics, Shanghai University, Shanghai 200444, China

Corresponding author: Zhang Da-Jun, djzhang@staff.shu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12271334, 12126352, 11875040)

文章全文 : translate this paragraph

参考文献

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[2]	Chen K, Zhang D J 2018 Appl. Math. Lett. 75 82 Google Scholar
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[4]	Ablowitz M J, Musslimani Z H 2014 Phys. Rev. E 90 032912
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施引文献

表 1 矩阵方程(26)的解

Table 1. Solutions to matrix equation (26)

No.	$(\sigma, \delta)$	${\boldsymbol T}$	${\boldsymbol A}$
1)	$(1, -1)$	$T_1=T_4=({\bf{0}})_N$, $T_3=-T_2 ={\boldsymbol{I} }_{N}$	$K_1=-K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
2)	$(1, 1)$	$T_1=T_4=({\bf{0}})_N$, $T_3=T_2 ={\boldsymbol{I} }_{N}$	$K_1=-K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
3)	$(-1, -1)$	$T_1=T_4=({\bf{0}})_N$, $T_3=T_2 ={\boldsymbol{I} }_{N}$	$K_1=K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
4)	$(-1, 1)$	$T_1=T_4=({\bf{0}})_N$, $T_3=-T_2 ={\boldsymbol{I} }_{N}$	$K_1=K^*_4={\boldsymbol{K} }_{N}\in \mathbb{C}_{N\times N}$
5)	$(-1, -1)$	$T_1=-T_4= {\boldsymbol{I} }_{N}, \; T_2=T_3 =({\bf{0} })_{N}$	$K_1={\boldsymbol{K} }_{N}\in \mathbb{R}_{N\times N}, \; K_4=-{\boldsymbol{H} }_{N}\in \mathbb{R}_{N\times N}$

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[1]	Chen K, Deng X, Lou S Y, Zhang D J 2018 Stud. Appl. Math. 141 113 Google Scholar
[2]	Chen K, Zhang D J 2018 Appl. Math. Lett. 75 82 Google Scholar
[3]	Ablowitz M J, Musslimani Z H 2013 Phys. Rev. Lett. 110 064105
[4]	Ablowitz M J, Musslimani Z H 2014 Phys. Rev. E 90 032912
[5]	Ablowitz M J, Musslimani Z H 2021 Phys. Lett. A 409 127516
[6]	Deng X, Lou S Y, Zhang D J 2018 Appl. Math. Comput. 332 477 Google Scholar
[7]	Chen K, Na C N, Yang J X 2023 Nonlinear Dyn. 111 1685 Google Scholar
[8]	Feng W, Zhao S L, Sun Y Y 2020 Int. J. Mod. Phys. B 34 2050021
[9]	Silem A, Wu H, Zhang D J 2021 Appl. Math. Lett. 116 107049
[10]	Chen K, Liu S M, Zhang D J 2019 Appl. Math. Lett. 88 230
[11]	Wang J, Wu H, Zhang D J 2020 Commun. Theor. Phys. 72 045002
[12]	Shi Y, Shen S F, Zhao S L 2019 Nonlinear Dyn. 95 1257 Google Scholar
[13]	Liu S Z, Wu H 2021 Mod. Phys. Lett. B 35 2150410
[14]	Wang J, Wu H 2022 Nonlinear Dyn. 109 3101 Google Scholar
[15]	Liu S M, Wu H, Zhang D J 2020 Rep. Math. Phys. 86 271 Google Scholar
[16]	Liu S Z, Wang J, Zhang D J 2022 Stud. Appl. Math. 148 651 Google Scholar
[17]	Wu H 2021 Nonlinear Dyn. 106 2497 Google Scholar
[18]	Liu S M, Wang J, Zhang D J 2022 Rep. Math. Phys. 89 199 Google Scholar
[19]	Wang J, Wu H, Zhang D J 2022 Chin. Phys. B 31 120201
[20]	Wang J, Wu H 2022 Commun. Nonlinear Sci. Numer. Simul. 104 106052
[21]	Zhang D J, Liu S M, Deng D 2023 Open Commun. Nonlinear Math. Phys. 3 23
[22]	Zhang D J 2020 Wronskian solutions of integrable systems, in Nonlinear Systems and Their Remarkable Mathematical Structures (Vol. 2) (Eds. Euler N, Nucci M C) (Boca Raton: CRC Press, Taylor & Francis) pp415–444
[23]	Hirota R 1974 Prog. Theore. Phys. 52 1498 Google Scholar
[24]	Freeman N C, Nimmo J J C 1983 Phys. Lett. A 95 1 Google Scholar
[25]	Nimmo J J C 1983 Phys. Lett. A 99 279 Google Scholar
[26]	Hietarinta J, Zhang D J 2009 J. Phys. A: Math. Theor. 42 404006
[27]	Ablowitz M J, Kaup D J, Newell A C, Segur H 1973 Phys. Rev. Lett. 31 125 Google Scholar
[28]	陈登远 2006 孤子引论 (北京: 科学出版社) Chen D Y 2006 Introduction to Soliton Theory (Beijing: Science Press) (in Chinese)
[29]	Newell A C 1985 Solitons in Mathematics and Physics (Philadelphin: SIAM)
[30]	Liu Q M 1990 J. Phys. Soc. Jpn. 59 3520 Google Scholar
[31]	Yin F M, Sun Y P, Cai F Q, Chen D Y 2008 Comm. Theore. Phys. 49 401 Google Scholar
[32]	Ablowitz M J, Ladik J F 1976 J. Math. Phys. 17 1011 Google Scholar
[33]	Fokas A S 1995 Physica D 87 145 Google Scholar
[34]	Lenells J, Fokas A S 2009 Nonlinearity 22 11 Google Scholar
[35]	Lenells J 2009 Stud. Appl. Math. 123 215 Google Scholar
[36]	Gerdjikov V S, Ivanov M I, Kulish P P 1980 Theor. Math. Phys. 44 784 Google Scholar
[37]	Zhang D J 2006 arXiv: nlin/0603008v3 [nlin.SI]
[38]	Zhang D J, Zhao S L, Sun Y Y, Zhou J 2014 Rev. Math. Phys. 26 1430006
[39]	Gürses M, Pekcan A 2018 J. Math. Phys. 59 051501
[40]	Gürses M, Pekcan A 2019 Commun. Nonlinear Sci. Numer. Simul. 71 161 Google Scholar
[41]	Gürses M, Pekcan A 2021 Commun. Nonlinear Sci. Numer. Simul. 97 105736

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