Symmetries and Exact Solutions of a 2+1-dimensional Sine-Gordon System

We investigate the classical and nonclassical reductions of the $2+1$-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a strong generalisation of the sine-Gordon equation. A family of solutions obtained as a nonclassical reduction involves a decoupled sum of solutions of a generalised, real, pumped Maxwell-Bloch system. This implies the existence of families of solutions, all occurring as a decoupled sum, expressible in terms of the second, third and fifth Painlev\'e transcendents, and the sine-Gordon equation. Indeed, hierarchies of such solutions are found, and explicit transformations connecting members of each hierarchy are given. By applying a known B\"acklund transformation for the system to the new solutions found, we obtain further families of exact solutions, including some which are expressed as the argument and modulus of sums of products of Bessel functions with arbitrary coefficients. Finally, we prove the sine-Gordon system has the Painlev\'e property, which requires the usual test to be modified, and derive a non-isospectral Lax pair for the generalised, real, pumped Maxwell-Bloch system.