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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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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
Xu D H, Lou S Y 2020 Acta Phys. Sin. 69 014208
Google Scholar
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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
[18] Li Y Q, Chen J C, Chen Y, Lou S Y 2014 Chin. Phys. Lett. 31 010201
Google Scholar
[19] 陈登远 2006 孤子引论 (北京: 科学出版社) pp14—42
Chen D Y 2006 Introduction on Solitons (Beijing: China Science Publishing and Media Ltd) pp14—42
[20] Lou S Y 2018 J. Math. Phys. 59 083507
Google Scholar
[21] Hietarinta J 1987 J. Math. Phys. 28 1732; 2094; 2586
[22] Chen K, Deng X, Lou S Y, Zhang D J 2018 Stud. Appl. Math. 141 113
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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