Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


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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