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 R1{{\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 CS2(1)↪C3{\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}.

Keywords

• coupled matrix nonlinear schrödinger-type equations
• complex grassmannian manifold
• schrödinger flow
• gauge equivalence
• soliton solutions
• 53c35
• 53a04
• 37k10
• 35q55
• 35c08