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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.
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Keywords:
- bilinearization-reduction approach /
- double Wronskian /
- integrable system /
- exact solution
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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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