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Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

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Abstract

Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.

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Authors and Affiliations

  1. Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesberg, South Africa

    F. M. Mahomed

  2. Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus of the College of Electrical and Mechanical Engineering, Peshawar Road, Rawalpindi, Pakistan

    Asghar Qadir

  3. Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Asghar Qadir

Authors
  1. F. M. Mahomed
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  2. Asghar Qadir
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Correspondence to Asghar Qadir.

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Mahomed, F.M., Qadir, A. Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations. Nonlinear Dyn 48, 417–422 (2007). https://doi.org/10.1007/s11071-006-9095-z

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  • DOI: https://doi.org/10.1007/s11071-006-9095-z

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