• P G L LEACH

      Articles written in Pramana – Journal of Physics

    • Cosmic strings in Bianchi III spacetime: Integrable cases

      S D Maharaj P G L Leach K S Govinder

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      We investigate the integrability of cosmic strings in Bianchi III spacetime using a symmetry analysis. The behaviour of the model is reduced to the solution of a single second order nonlinear differential equation. We show that this equation has a rich structure and admits an infinite family of solutions. Our class of solutions extends special cases previously obtained by Tikekar and Patel [Gen. Relativ. Gravit.24, 397 (1992)].

    • Cosmic strings in Bianchi II, VIII and IX spacetimes: Integrable cases

      L K Patel S D Maharaj P G L Leach

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      We investigate the integrability of cosmic strings in Bianchi II, VIII and IX space-times using a Lie symmetry analysis. The behaviour of the gravitational field is governed by solutions of a single second order nonlinear differential equation. We demonstrate that this equation is integrable and admits an infinite family of physically reasonable solutions. Particular solutions obtained by other authors are shown to be special cases of our class of solutions.

    • Algebraic resolution of the Burgers equation with a forcing term

      R SINUVASAN K M TAMIZHMANI P G L LEACH

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      We introduce an inhomogeneous term, $f (t,x)$, into the right-hand side of the usual Burgers equation and examine the resulting equation for those functions which admit at least one Lie point symmetry. For those functions $f (t,x)$ which depend nontrivially on both $t$ and $x$, we find that there is just one symmetry. If $f$ is a function of only $x$, there are three symmetries with the algebra $sl(2,R)$. When $f$ is a function of only $t$ , there are five symmetries with the algebra $sl(2,R)\oplus_{s} 2A_1$. In all the cases, the Burgers equation is reduced to the equation for a linear oscillator with nonconstant coefficient.

    • Symmetries and integrability of the modified Camassa–Holm equation with an arbitrary parameter

      A DURGA DEVI K KRISHNAKUMAR R SINUVASAN P G L LEACH

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      We study the symmetry and integrability of a modified Camassa–Holm equation (MCH), with an arbitrary parameter $k$, of the form $u_t + k(u − u_{xx})^{2}u_x − u_{xxt} + (u^2 − u_x^2)(u_x − u_{xxx} ) = 0.$ The commutator table and adjoint representation of the symmetries are presented to construct one-dimensional optimal system. By using the one-dimensional optimal system, we reduce the order or number of independent variables of the above equation and also we obtain interesting novel solutions for the reduced ordinary differential equations. Finally, we apply the Painlevé test to the resultant nonlinear ordinary differential equation and it is observed that the equation is integrable.

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