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There are three di erent actions of the unimodular Lie group SL on a two dimensional space In every case we show how an ordinary di erential equation admitting SL as a symmetry group can be reduced in order by three and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation A… (More)

In this paper we describe Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation (PIV) dw dz = 1 2w ( dw dz )2 + 3 2 w + 4zw + 2(z − α)w + β w , (1) with α, β constants. Specifically, a nonlinear superposition principle for PIV, hierarchies of solutions expressible in terms of complementary error or parabolic cylinder… (More)

- Peter A. Clarkson, Elizabeth L. Mansfield
- SIAM Journal of Applied Mathematics
- 1994

The second Painlevé hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlevé equation, PII . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlevé… (More)

- Elizabeth L. Mansfield, Peter A. Clarkson
- J. Symb. Comput.
- 1997

We show how the maple package diffgrob2 can be used to analyse overdetermined systems of pde. The particular application discussed here is to find classical symmetries of differential equations of mathematical and physical interest. Symmetries of differential equations underly most of the methods of exact integration known; the use and calculation of such… (More)

The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic… (More)

We investigate the classical and nonclassical reductions of the 2+1-dimensional sine-Gordon system of Konopelchenko and Rogers, which is a strong generalisation of the sine-Gordon equation. A family of solutions obtained as a nonclassical reduction involves a decoupled sum of solutions of a generalised, real, pumped Maxwell-Bloch system. This implies the… (More)

In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along… (More)

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but… (More)

In this paper we study symmetry reductions of a class of nonlinear third order partial differential equations ut − ǫuxxt + 2κux = uuxxx + αuux + βuxuxx , (1) where ǫ, κ, α and β are arbitrary constants. Three special cases of equation (1) have appeared in the literature, up to some rescalings. In each case the equation has admitted unusual travelling wave… (More)