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中国物理学会期刊

互反型高维可积Kaup-Newell系统

CSTR: 32037.14.aps.72.20222418

Higher dimensional reciprocal integrable Kaup-Newell systems

CSTR: 32037.14.aps.72.20222418
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  • 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.

     

    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.

     

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