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

In this article, we prove that the equation of the Schr"odinger maps from ${\mathbf{R}}^2$ to the hyperbolic 2-space ${\mathbf{H}}^2$ 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