Explicit Blowing-up Solutions to the Schr\"odinger Maps from ${\bf R}^2$ to the Hyperbolic 2-Space ${\bf H}^2$

Abstract

In this article, we prove that the equation of the Schr\"odinger maps from ${\bf R}^2$ to the hyperbolic 2-space ${\bf H}^2$ is SU(1,1)-gauge equivalent to the following 1+2 dimensional nonlinear Schr\"odinger-type system of unknown three complex functions $p,q,r$ and a real function $u$: {c} iq_t+q_{z{\bar z}}-2u q+2({\bar p}q)_z-2pq_{\bar z}-4|p|^2q=0\qquad ir_t-r_{z{\bar z}}+2u r+2({\bar p}r)_z-2pr_{\bar z}+4|p|^2r=0\qquad ip_t+(qr)_{\bar z}-u_z=0\qquad\qquad\qquad\qquad\qquad\qquad