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## Higher dimensional reciprocal integrable Kaup-Newell systems

Lou Sen-Yue, Hao Xia-Zhi, Jia Man
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• #### 摘要

可积系统研究是物理和数学等学科的重要研究课题. 然而, 通常的可积系统研究往往被限制在(1+1)维和(2+1)维, 其原因是高维可积系统极其稀少. 最近, 我们发现利用形变术可以从低维可积系统导出大量的高维可积系统. 本文利用形变术, 将(1+1)维的Kaup-Newell (KN)系统推广到(4+1)维系统. 新系统除了包含原来的(1+1)维的KN系统外, 还包含三种(1+1)维KN系统的互反形式. 模型也包含了许多新的(D+1)维($D\leqslant3$)的互反型可积系统. (4+1)维互反型KN系统的Lax可积性和对称可积性也被证明. 新的互反型高维KN系统的求解非常困难. 本文仅研究(2+1)维互反型导数非线性薛定谔方程的行波解, 并给出薛定谔方程孤子解的隐函数表达式.

#### Abstract

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.

#### 作者及机构信息

###### 通信作者: 楼森岳, lousenyue@nbu.edu.cn
• 基金项目: 国家自然科学基金(批准号: 12235007, 11975131, 11435005)、宁波大学王宽诚幸福基金和浙江省自然科学基金(批准号: LQ20A010009)资助的课题

#### Authors and contacts

###### Corresponding author: Lou Sen-Yue, lousenyue@nbu.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12235007, 11975131, 11435005), the K. C. Wong Magna Fund in Ningbo University, and the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ20A010009)

#### 施引文献

• 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

Fig. 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)