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The study of integrable systems is one of important topics both in physics and in mathematics. However, traditional studies on integrable systems are usually restricted in (1+1) and (2+1) dimensions. The main reasons come from the fact that high-dimensional integrable systems are extremely rare. Recently, we found that a large number of high dimensional integrable systems can be derived from low dimensional ones by means of a deformation algorithm. In this paper, the (1+1) dimensional Kaup-Newell (KN) system is extended to a (4+1) dimensional system with the help of the deformation algorithm. In addition to the original (1+1) dimensional KN system, the new system also contains three reciprocal forms of the (1+1) dimensional KN system. The model also contains a large number of new (D+1) dimensional (
$D \leqslant 3$ ) integrable systems. The Lax integrability and symmetry integrability of the (4+1) dimensional KN system are also proved. It is very difficult to solve the new high-dimensional KN systems. In this paper, we only investigate the traveling wave solutions of a (2+1) dimensional reciprocal derivative nonlinear Schrödinger equation. The general envelope travelling wave can be expressed by a complicated elliptic integral. The single envelope dark (gray) soliton of the derivative nonlinear Schödinger equation can be implicitly written.-
Keywords:
- higher dimensional integrable models /
- Kaup-Newell systems /
- deformation algorithm /
- travelling wave solutions
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图 1 导数非线性薛定谔方程(56)的单孤子解((61)式) (a) 参数取值为
$ b = 20, \ c = 1, \ k_2 = 1/2000, \ k_1 = 2, \ a = 1, \ X_0 = 0 $ ; (b) 参数取值为$ b = 20, \ c = 1, \ k_2 = 1, \ k_1 = 2, \ a = 1, \ X_0 = 1.563 $ ; (c) 由(61)式和$ c = 0 $ 表示的暗尖峰孤子结构, 其他参数与图(b)相同Figure 1. Single soliton solution (Eq. (61)) of the nonlinear Schrödinger equation (56): (a) Parameters are selected as
$b = 20, $ $ \ c = 1, \ k_2 = 1/2000, \ k_1 = 2, \ a = 1, \ X_0 = 0$ ; (b) parameters are selected as$b = 20, \ c = 1, \ k_2 = 1, \ k_1 = 2, $ $ \ a = 1, \ X_0 = 1.563$ ; (c) a dark peakon soliton solution expressed by Eq. (61) with$ c = 0 $ and other parameters are same as in panel (b) -
[1] Gardner C S, Greene J M, Kruskal M D, Miura R M 1967 Phys. Rev. Lett. 19 1095
Google Scholar
[2] Miura R M 1968 J. Math. Phys. 9 1022
[3] Olver P J 1993 Applications of Lie Groups to Differential Equations (2nd Ed.) (New York: Springer)
[4] Lou S Y 1993 Phys. Rev. Lett. 71 4099
Google Scholar
[5] Lou S Y, Hu X B, Liu Q P 2021 JHEP 07 058
Google Scholar
[6] Ramani A, Gramaticos B, Bountis T 1989 Phys. Rep. 180 159
Google Scholar
[7] Conte R 1989 Phys. Lett. A 140 383
Google Scholar
[8] Lou S Y 1998 Z. Naturforsch 53a 251
[9] 楼森岳 1998 物理学报 47 1937
Google Scholar
Lou S Y 1998 Acta Phys. Sin. 47 1937
Google Scholar
[10] Lou S Y 1998 Phys. Rev. Lett. 80 5027
Google Scholar
[11] Hirota R 1971 Phys. Rev. Lett. 27 1192
Google Scholar
[12] Rogers C, Schief W K 2002 Backlund and Darboux Transformations, Geometry and Modern Applications in Soliton Theory (Cambridge: Cambridge University Press)
[13] Liu Q P, Mañas M 2000 Phys. Lett. B 485 293
Google Scholar
[14] Hao X Z, Lou S Y 2022 Math. Meth. Appl. Sci. 45 5774
Google Scholar
[15] Lou S Y 1993 Phys. Lett. B 302 261
Google Scholar
[16] Nijhoff F W, Sun Y Y, Zhang D J 2023 Commun. Math. Phys. 399 599
Google Scholar
[17] Zhang D J, Zhao S L 2013 Stud. Appl. Math. 131 72
Google Scholar
[18] Liu Q P, Hu X B 2005 J. Phys. A 38 6371
Google Scholar
[19] Gao X N, Lou S Y, Tang X Y 2013 JHEP 05 029
Google Scholar
[20] Xia S Q, Kaltsas D, Song D H, et al. 2021 Science 372 72
Google Scholar
[21] Loutsenko I, Roubtsov D 1997 Phys. Rev. Lett. 78 3011
Google Scholar
[22] Weigel H, Gamberg L, Reinhardt H 1997 Phys. Rev. D 55 6910
Google Scholar
[23] Dolan L 1997 Nucl. Phys. B 489 245
Google Scholar
[24] Chiueh T, Woo T P 1997 Phys. Rev. E 55 1048
Google Scholar
[25] Tajiri M, Maesono H 1997 Phys. Rev. E 55 3351
[26] Chang D E, Vuletic V, Lukin M D 2014 Nat. Photonics 8 685
Google Scholar
[27] Das G C 1997 Phys. Plasmas 4 2095
Google Scholar
[28] Jia M, Lou S Y 2006 Phy. Lett. A 353 407
Google Scholar
[29] Hu H C, Lou S Y, Chow K W 2007 Chaos, Solitons and Fractals 31 1213
Google Scholar
[30] Lou S Y 1997 J. Phys. A: Math. Phys. 30 7259
Google Scholar
[31] Lou S Y, Hao X Z, Jia M 2023 JHEP 03 018
[32] Kaup D J, Newell A C 1978 J. Math. Phys. 19 798
Google Scholar
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