an:05236977
Zbl 1132.35328
Harris, Shirley E.; Clarkson, Peter A.
Painlev?? analysis and similarity reductions for the magma equation
EN
SIGMA, Symmetry Integrability Geom. Methods Appl. 2, Paper 068, 17 p. (2006).
00181885
2006
j
35C05 35Q58 37K10
Painlev?? analysis; similarity reductions; magma equation
Summary: 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 Painlev?? 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 \(m=\frac 12, n = -\frac 12\), it is a real, generalized, pumped Maxwell-Bloch equation.