Explicit blow-up solutions to the Schrodinger maps from R2 to the hyperbolic 2-space H2

In this article, we prove that the equation of the Schrodinger maps from R-2 to the hyperbolic 2-space H-2 is SU (1,1)-gauge equivalent to the following 1+ 2 dimensional nonlinear Schrodinger-type system of three unknown complex functions p, q, r, and a real function u: iq(t) + q(z (z) over bar)-2uq+2((p) over barq)(z)-2pq((z) over bar)-4 vertical bar p vertical bar(2)q=0, ir(t)-r(z (z) over bar)+2ur +2((p) over barr)(z)-2pr((z) over bar)+ 4 vertical bar p vertical bar(2)r=0, ip(t)+(qr)((z) over bar)-u(z)=0, (P) over bar (z)+p((z) over bar)-vertical bar vertical bar(2)+vertical bar r vertical bar(2),-(r) over bar (z)+q ($) over bar -2(p (r) over bar +(p) over barq) , where z is a complex coordinate of the plane R-2 and (z) over bar is the complex conjugate of z. Although this nonlinear Schrodinger-type system looks complicated, it admits a class of explicit blow-up smooth solutions: p=0, q=(e(i(bz (z) over bar /2(a+bt))/a +bt)alpha(z) over bar= r-e(-i(bz (z) over bar /2(a+bt)))/(a+bt)alpha(z) over bar, u=2 alpha(2)z (z) over bar/(a+bt)(2), where a and b are real numbers with ab < 0 and alpha satisfies alpha(2)= b(2)/16. From these facts, we explicitly construct smooth solutions to the Schrodinger maps from R-2 to the hyperbolic 2-space H-2 by using the gauge transformations such that the absolute values of their gradients blow up in finite time. This reveals some blow- up phenomenon of Schrodinger maps. (C) 2009 American Institute of Physics. [doi:10.1063/1.3218848]