Geometric realizations of Fordy–Kulish nonlinear Schrödinger systems
Geometric realizations of Fordy–Kulish nonlinear Schrödinger systems
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DOI：
10.2140/pjm.2000.195.157
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发表时间：
2000-09
影响因子：
0.6
通讯作者：
Joel Langer;R. Perline
中科院分区：
文献类型：
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作者：
Joel Langer;R. Perline
将Sym和Pohlmeyer方法应用于与厄米对称空间相关的Fordy-Kulish广义非线性薛定谔系统，得到了许多可积系统的几何实现。所得几何方程对应于在李代数中演化的不同弧长参数化曲线，推广了涡旋丝运动的局部感应模型。建立了这类曲线的自然Frenet理论，并用几何递归算子分析了曲线演化与自然曲率演化的一般对应关系。对对称空间SO(p + 2)/SO(p) × SO(2)进行适当的专门化，得到Rp+1和Sp中自然曲率满足广义mKdV系统的曲线的演化方程。这个例子与Doliwa和Santini最近的建筑有关，并阐明了后者的某些特征。∗部门。数学，凯斯西储大学，克利夫兰OH 44106电子邮件：jxl6@po.cwru.edu†系。数学与计算科学数学学科分类：58F07, 35Q55
A method of Sym and Pohlmeyer, which produces geometric realizations of many integrable systems, is applied to the Fordy-Kulish generalized non-linear Schrodinger systems associated with Hermitian symmetric spaces. The resulting geometric equations correspond to distinguished arclength-parametrized curves evolving in a Lie algebra, generalizing the localized induction model of vortex filament motion. A natural Frenet theory for such curves is formulated, and the general correspondence between curve evolution and natural curvature evolution is analyzed by means of a geometric recursion operator. An appropriate specialization in the context of the symmetric space SO(p + 2)/SO(p) × SO(2) yields evolution equations for curves in Rp+1 and Sp, with natural curvatures satisfying a generalized mKdV system. This example is related to recent constructions of Doliwa and Santini and illuminates certain features of the latter. ∗Dept. of Mathematics, Case Western Reserve University, Cleveland OH 44106 email: jxl6@po.cwru.edu †Dept. of Mathematics and Comp. Sci., Drexel University, Philadelphia PA 19104 email: rperline@mcs.drexel.edu Mathematics Subject Classification: 58F07, 35Q55