Abstract
The Lie and Noether point symmetry analyses of a kth-order system of m complex ordinary differential equations (ODEs) with m dependent variables are performed. The decomposition of complex symmetries of the given system of complex ODEs yields Lie- and Noether-like operators. The system of complex ODEs can be split into 2m coupled real partial differential equations (PDEs) and 2m Cauchy–Riemann (CR) equations. The classical approach is invoked to compute the symmetries of the 4m real PDEs and these are compared with the decomposed Lie- and Noether-like operators of the system of complex ODEs. It is shown that, in general, the Lie- and Noether-like operators of the system of complex ODEs and the symmetries of the decomposed system of real PDEs are not the same. A similar analysis is carried out for restricted systems of complex ODEs that split into 2m coupled real ODEs. We summarize our findings on restricted complex ODEs in two propositions.
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Acknowledgements
FMM thanks the NRF of South Africa for a research grant. RN is thankful to Lahore school of Economics for a research grant. The authors are also grateful to the referee for suggesting points that have improved this paper.
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NAZ, R., MAHOMED, F.M. Lie and Noether symmetries of systems of complex ordinary differential equations and their split systems. Pramana - J Phys 83, 9–20 (2014). https://doi.org/10.1007/s12043-014-0762-1
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DOI: https://doi.org/10.1007/s12043-014-0762-1