Movable algebraic singularities of second-order ordinary differential equations (2008)
Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the...
Tropical Nevanlinna theory and ultra-discrete equations (2007)
Halburd, R. G., Southall, N. J.
A tropical version of Nevanlinna theory is described in which the role of meromorphic functions is played by continuous piecewise linear functions of a real variable whose one-sided derivatives are...
Finite-order meromorphic solutions and the discrete Painleve equations (2007)
Halburd, R. G., Korhonen, R. J.
Let w(z) be an admissible finite-order meromorphic solution of the second-order difference equation where R(z, w(z)) is rational in w(z) with coefficients that are meromorphic in z. Then either w(z)...
The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations...
The C-metric as a colliding plane wave space-time (2007)
Griffiths, J. B., Halburd, R. G.
It is explicitly shown that part of the C-metric space-time inside the black hole horizon may be interpreted as the interaction region of two colliding plane waves with aligned linear polarization,...
The C-metric as a colliding plane wave space-time (2007)
Griffiths, J.B., Halburd, R.G.
This is a pre-print.
The C-metric as a colliding plane wave space-time (2007)
Griffiths, J.B., Halburd, R.G.
This is a pre-print.
Finite-order meromorphic solutions and the discrete Painleve equations (2006)
Let w(z) be an admissible finite-order meromorphic solution of the second-order difference equation w(z+1) + w(z-1) = R(z,w(z)) where R(z, w(z)) is rational in w(z) with coefficients that are...
Barnett, D.C., Halburd, R.G., Korhonen, R.J., Morgan, W.
This is a pre-print.
Barnett, D.C., Halburd, R.G., Korhonen, R.J., Morgan, W.
This is a pre-print.
Finite-order meromorphic solutions and the discrete Painleve equations (2006)
Halburd, R. G., Korhonen, R. J.
Let w(z) be an admissible finite-order meromorphic solution of the second-order difference equation\[ w(z+1)+w(z-1) = R(z,w(z)) \]where R(z, w(z)) is rational in w(z) with coefficients that are...
Nevanlinna theory for the difference operator (2005)
Halburd, R. G., Korhonen, R. J.
Certain estimates involving the derivative $f\mapsto f'$ of a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to...
Finite-order meromorphic solutions and the discrete Painleve equations (2005)
Halburd, R. G., Korhonen, R. J.
Let w(z) be a finite-order meromorphic solution of the second-order difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is rational in w(z) and meromorphic in z. Then either w(z)...
Diophantine Integrability (2005)
The heights of iterates of the discrete Painleve equations over number fields appear to grow no faster than polynomials while the heights of generic solutions of non-integrable discrete equations...
Halburd, R. G., Korhonen, R. J.
The Lemma on the Logarithmic Derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary differential equations. In this paper, a difference analogue...
Nevanlinna theory for the difference operator (2005)
This is a pre-print.
Diophantine integrability (2005)
This pre-print has been submitted and accepted to the journal, Journal of Physics A - Mathematical and General. The definitive version: HALBURD, R.G., 2005. Diophantine integrability. Journal of...
This is a pre-print.
Finite-order meromorphic solutions and the discrete painleve equations (2005)
This is a pre-print.
Nevanlinna theory for the difference operator (2005)
This is a pre-print.
Diophantine integrability (2005)
This pre-print has been submitted and accepted to the journal, Journal of Physics A - Mathematical and General. The definitive version: HALBURD, R.G., 2005. Diophantine integrability. Journal of...
This is a pre-print.
Finite-order meromorphic solutions and the discrete painleve equations (2005)
This is a pre-print.
Integrable systems and reductions of the self-dual Yang–Mills equations (2003)
Ablowitz, M.J., Chakravarty, S., Halburd, R.G.
Many integrable equations are known to be reductions of the self-dual Yang–Mills equations. This article discusses some of the well known reductions including the standard soliton equations, the...
On the meromorphic solutions to an equation of Hayman (2002)
This pre-print has been submitted, and accepted, to the journal, Journal of Mathematical Analysis and Applications [© Elsevier]. The definitive version: CHIANG, Y.M. and HALBURD, R.G., 2002. On the...
First integrals of a generalized Darboux-Halphen system. (2002)
Chakravarty, S., Halburd, R.G.
This pre-print has been submitted, and accepted, to the journal, Journal of Mathematical Physics [© American Institute of Physics]. The definitive version: HALBURD, R. and CHAKRAVARTY, S., 2003....
Self-similar solutions of certain coupled integrable systems (2002)
Chakravarty, S., Halburd, R.G., Kent, S.L.
This pre-print has been submitted, and accepted, to the journal, Journal of Physics A - Mathematical and General [© Institute of Physics]. The definitive version: CHAKRAVARTY, S., HALBURD, R.G. and...
Shear-free relativistic fluids and the absence of movable branch points (2002)
This pre-print has been submitted, and accepted, to the journal, Journal of Mathematical Physics [© American Institute of Physics]. The definitive version: HALBURD, R.G., 2002. Shear-free...
On the meromorphic solutions to an equation of Hayman (2002)
This pre-print has been submitted, and accepted, to the journal, Journal of Mathematical Analysis and Applications [© Elsevier]. The definitive version: CHIANG, Y.M. and HALBURD, R.G., 2002. On the...
First integrals of a generalized Darboux-Halphen system. (2002)
Chakravarty, S., Halburd, R.G.
This pre-print has been submitted, and accepted, to the journal, Journal of Mathematical Physics [© American Institute of Physics]. The definitive version: HALBURD, R. and CHAKRAVARTY, S., 2003....
Self-similar solutions of certain coupled integrable systems (2002)
Chakravarty, S., Halburd, R.G., Kent, S.L.
This pre-print has been submitted, and accepted, to the journal, Journal of Physics A - Mathematical and General [© Institute of Physics]. The definitive version: CHAKRAVARTY, S., HALBURD, R.G. and...
Shear-free relativistic fluids and the absence of movable branch points (2002)
This pre-print has been submitted, and accepted, to the journal, Journal of Mathematical Physics [© American Institute of Physics]. The definitive version: HALBURD, R.G., 2002. Shear-free...
Solvable models of relativistic charge fluid spheres (2000)
This is a pre-print. The definitive version: HALBURD, R., 2001. Solvable models of relativistic charge fluid spheres. Classical and Quantum Gravity, 18(1), pp. 11-25.
Solvable models of relativistic charge fluid spheres (2000)
This is a pre-print. The definitive version: HALBURD, R., 2001. Solvable models of relativistic charge fluid spheres. Classical and Quantum Gravity, 18(1), pp. 11-25.