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多孤子解是非线性数学物理系统的基本激发模式. 文献中存在各种类型的表达式, 如广田(Hirota)形式, 朗斯基(Wronskian)或双朗斯基形式和法夫(Phaffian)形式. 最近在多地系统的研究中, 我们发现使用一种全新但等价的形式具有极为简洁和方便的优点. 本文主要综述多种类型可积非线性系统的多孤子解的新型表达式, 同时对SK方程、非对称NNV系统、修正KdV型、sG型、AKNS模型和全离散H1系统也给出一些文献中还没出现过的新的更为简便的表达式. 新的孤子表达式通常具有显然的时空全反演(包括时间反演、空间反演、孤子初始位置反演及电荷共轭反演(正反粒子反演))对称性. 这种具有显式全反演对称性的表达式在研究多地非局域系统和局域和非局域可积系统的各种共振结构时具有很大的优越性.Multiple soliton solutions are fundamental excitations. There are many kinds of equivalent representations for multiple soliton solutions such as the Hirota forms, Wronskian and/or double Wronskian expressions and Phaffian representations. Recently, in the studies of multi-place nonlocal systems, we find that there are a type of novel but equivalent simple and elegant forms to describe multiple soliton solutions for various integrable systems. In this paper, we mainly review novel types of expressions of multiple soliton solutions for some kinds of nonlinear integrable systems. Meanwhile, some completely new expressions for the Sawada-Kortera equations, the asymmetric Nizhnik-Novikov-Veselov system, the modified KdV equation, the sine-Gordon equation, the Ablowitz-Kaup-Newell-Segue system and the completely discrete H1 equation are firstly given in this paper. New expressions usually possess explicit full reversal symmetries including parity, time reversal, soliton initial position reversal and charge conjugate reversal. These kinds of explicitly symmetric forms are very useful and convenient in the studies on the nonlinear physical problems such as the multi-place nonlocal systems and the resonant structures.
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Keywords:
- Integrable systems /
- multiple soliton solutions /
- full reversal symmetries /
- multi-place systems
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[1] Russell J S 1837 Rep. Meet. Brit. Assoc. Adv. Sci. 7th 417
[2] Zabusky N J, Kruskal M D 1965 Phys. Rev. Lett. 15 240
[3] Gardner C S, Greene J M, Kruskal M D, Miura R M 1976 Phys. Rev. Lett. 19 1095
[4] Kivshar Y S, Malomed B A 1989 Rev. Mod. Phys. 61 763
Google Scholar
[5] Köttig F, Tani T, Travers J C, Russell P St J 2017 Phys. Rev. Lett. 118 263902
Google Scholar
[6] Wright L G, Christodoulides D N, Wise F W 2017 Science 358 94
Google Scholar
[7] Dudley J M, Dias F, Erkintalo M, Genty G 2014 Nat. Photonics 8 755
Google Scholar
[8] Stratmann M, Pagel T, Mitschke F 2005 Phys. Rev. Lett. 95 143902
Google Scholar
[9] Herink G, Kurtz F, Jalali B, Solli D R, Ropers C 2017 Science, 356 50
Google Scholar
[10] Liu X M, Yao X K, Cui Y D 2018 Phys. Rev. Lett. 121 023905
Google Scholar
[11] Strogatz S 2001 Nature (London) 410 268
Google Scholar
[12] Forte S 1992 Rev. Mod. Phys. 64 193
Google Scholar
[13] Hertog T, Horowitz G T 2005 Phys. Rev. Lett. 94 221301
Google Scholar
[14] Drummond P D, Kheruntsyan K V, He H 1998 Phys. Rev. Lett. 81 3055
Google Scholar
[15] Lou S Y, Huang F 2017 Sci. Rep. 7 869
Google Scholar
[16] Hirota R 2004 The Direct Method in Soliton Theory, Edited and translated by Nagai A, Nimmo J, Gilson C, Cambridge Tracts in Mathematics No. 155 (Cambrifge: Cambridge University Press) pp1−61
[17] Gu C H, Hu H S, Zhou Z X 2005 Darboux Transformations in Integrable Systems: Theory and their Applications to Geommetry (Dordrecht, Netherland: Springer) pp1–64
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Google Scholar
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Chen D Y 2006 Introduction on Solitons (Beijing: China Science Publishing and Media Ltd) pp14—42
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Google Scholar
[21] Hietarinta J 1987 J. Math. Phys. 28 1732; 2094; 2586
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Google Scholar
[23] Ablowitz M J, Musslimani Z H 2016 Nonlinearity 29 915
Google Scholar
[24] Ablowitz M J, Kaup D J, Newell A C, Segur H 1974 53 249
[25] Hietarinta J, Zhang D J 2009 J. Phys. A: Math. Theor. 42 404006
Google Scholar
[26] Lou S Y 2019 Stud. Appl. Math. 143 123; 2018 arXiv: 1806.07559[nlin.SI]
[27] Li C C, Lou S Y, Jia M 2018 Nonl. Dynamics, 93 1799
Google Scholar
[28] 徐丹红, 楼森岳 2020 物理学报 69 014208
Google Scholar
Xu D H, Lou S Y 2020 Acta Phys. Sin. 69 014208
Google Scholar
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