On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

• Published in: Communications in theoretical physics Vol. 45; no. 1; pp. 135 - 142
• Main Authors: Jun-Yi, Wu, Qing, Ding, Tenenblat, Keti
• Format: Journal Article
• Language: English
• Published: IOP Publishing 15.01.2006; Key Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 200433, China%Department of Mathematics, Brasilia University, Brasilia DF 70910-900, Brazil; Institute of Mathematics, Fudan University, Shanghai 200433, China%Institute of Mathematics, Fudan University, Shanghai 200433, China
• Subjects: 三维双曲线空间; 守恒法则; 微分方程; 非线性Schrodinger方程; conservation laws; differential equations describing 3-dimensional hyperbolic spaces; (2+1)-dimensional integrable systems
• ISSN: 0253-6102
• DOI: 10.1088/0253-6102/45/1/026

In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.