Exact vortex solutions of the complex sine-Gordon theory on the plane
  We construct explicit multivortex solutions for the first and second complex sine-Gordon equations. The constructed solutions are expressible in terms of the modified Bessel and rational functions, respectively. The vorticity-raising and lowering Backlund transformations are interpreted as the Schlesinger transformations of the fifth Painleve equation. 
I.V. Barashenkov and D.E. Pelinovsky