Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow

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} .