Exact vortex solutions of the complex sine-Gordon theory on the plane

TL;DR

This paper constructs explicit multivortex solutions for the complex sine-Gordon equations, expressing them with special functions and linking transformations to Painleve equations.

Contribution

It provides explicit multivortex solutions for the complex sine-Gordon equations and interprets Bäcklund transformations as Schlesinger transformations.

Findings

Solutions expressed via modified Bessel and rational functions

Bäcklund transformations linked to Painleve V transformations

Explicit multivortex solutions on the plane

Abstract

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.