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We investigate the classical and nonclassical reductions of the$2+1$-dimensional sine-Gordon system of Konopelchenko and Rogers, which is astrong generalisation of the sine-Gordon equation. A family of solutionsobtained as a nonclassical reduction involves a decoupled sum of solutions of ageneralised, real, pumped Maxwell-Bloch system. This implies the existence offamilies of solutions, all occurring as a decoupled sum, expressible in termsof the second, third and fifth Painlev\'e transcendents, and the sine-Gordonequation. Indeed, hierarchies of such solutions are found, and explicittransformations connecting members of each hierarchy are given. By applying aknown B\"acklund transformation for the system to the new solutions found, weobtain further families of exact solutions, including some which are expressedas the argument and modulus of sums of products of Bessel functions witharbitrary coefficients. Finally, we prove the sine-Gordon system has thePainlev\'e property, which requires the usual test to be modified, and derive anon-isospectral Lax pair for the generalised, real, pumped Maxwell-Blochsystem.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system