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 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 ≤ 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 expressed by a complicated elliptic integral. The single envelope dark (gray) soliton of the derivative nonlinear Schodinger equation can be implicitly written.