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SIGMA 2 (2006), 068, 17 pages nlin.SI/0610011
https://doi.org/10.3842/SIGMA.2006.068
Painlevé Analysis and Similarity Reductions for the Magma Equation
Shirley E. Harris a and Peter A. Clarkson b
a) Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 3LB, UK
b) Institute of Mathematics, Statistics and Actuarial Science,
University of Kent, Canterbury, CT2 7NF, UK
Received September 27, 2006; Published online October 05, 2006
Abstract
In this paper, we examine a generalized magma equation for rational
values of two parameters, m and n. Firstly, the similarity reductions are
found using the Lie group method of infinitesimal transformations. The Painlevé ODE test
is then applied to the travelling wave reduction, and the pairs of m and n which pass the
test are identified. These particular pairs are further subjected to the ODE test on their other
symmetry reductions. Only two cases remain which pass the ODE test for all such symmetry reductions
and these are completely integrable. The case when m = 0, n = −1 is related to the Hirota-Satsuma
equation and for m = ½, n = −½, it is a real, generalized,
pumped Maxwell-Bloch equation.
Key words:
Painlevé analysis; similarity reductions; magma equation.
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