Affiliation:
1. School of Mathematics and Computer Sciences, Gannan Normal University , Ganzhou 341000 , P. R. China
Abstract
Abstract
In this article, the coupled matrix nonlinear Schrödinger (NLS) type equations are gauge equivalent to the equation of Schrödinger flow from
R
1
{{\mathbb{R}}}^{1}
to complex Grassmannian manifold
G
˜
n
,
k
=
GL
(
n
,
C
)
∕
GL
(
k
,
C
)
×
GL
(
n
−
k
,
C
)
,
{\widetilde{G}}_{n,k}={\rm{GL}}\left(n,{\mathbb{C}})/{\rm{GL}}\left(k,{\mathbb{C}})\times {\rm{GL}}\left(n-k,{\mathbb{C}}),
which generalizes the correspondence between Schrödinger flow to the complex 2-sphere
C
S
2
(
1
)
↪
C
3
{\mathbb{C}}{{\mathbb{S}}}^{2}\left(1)\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\mathbb{C}}}^{3}
and the coupled Landau-Lifshitz (CLL) equation. This gives a geometric interpretation of the matrix generalization of the coupled NLS equation (i.e., CLL equation) via Schrödinger flow to the complex Grassmannian manifold
G
˜
n
,
k
{\widetilde{G}}_{n,k}
. Finally, we explicit soliton solutions of the Schrödinger flow to the complex Grassmannian manifold
G
˜
2
,
1
{\widetilde{G}}_{2,1}
.