Clarkson, P.A. and Mansfield, E.L. and Milne, A.E. (1996) Symmetries and exact solutions of a (2+1)-dimensional sine-Gordon system. Philosophical Transactions of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences, 354 (1713). pp. 1807-1835.
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We investigate the classical and non-classical reductions of the (2 + 1)-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a strong generalization of the sine-Gordon equation. A family of solutions obtained as a non-classical reduction involves a decoupled sum of solutions of a generalized, 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 Painleve 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 Backlund transformation for the system to the new solutions found, rye 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 show that the sine-Gordon system satisfies the necessary conditions of the Painleve PDE test due to Weiss et al., which requires the usual test to be modified, and derive a non-isospectral Lax pair for the generalized, real, pumped Maxwell-Bloch system.
|Divisions:||Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science|
|Depositing User:||M.A. Ziai|
|Date Deposited:||29 Jun 2011 09:20|
|Last Modified:||29 Jun 2011 09:20|
|Resource URI:||http://kar.kent.ac.uk/id/eprint/18878 (The current URI for this page, for reference purposes)|
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