Higher dimensional reciprocal integrable Kaup-Newell systems

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 (<i>D</i>+1) dimensional (<inline-formula><tex-math id="M2">\begin{document}$D \leqslant 3$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222418_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20222418_M2.png"/></alternatives></inline-formula>) 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.