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Complex Lie symmetries for scalar second-order ordinary differential equations. (English) Zbl 1187.34044

The symmetry analysis of differential equations, started by Sophus Lie in XIX Century (and by now a standard tool in the study of nonlinear problems) is based on the action of smooth vector fields. When one considers complex equations, however, it is rather natural to require analyticity. Based on this remark, the authors study complex ODEs (CODEs) of second order and their symmetry properties. A CODE is equivalent to a system of two real ODEs; the analyticity requirement, which can be expressed through the Cauchy-Riemann condition, adds two real PDEs, so that we are faced with a mixed system of ODEs and PDEs (as well known, the presence of the latter makes actually easier to determine explicitly the symmetries).
Moreover, a system of two ODEs can be obtained from a CODE by restricting the complex function to a single real variable (the authors suggest the name r-CODE for this case). An analysis of complex Lie symmetries of r-CODEs yields real Lie symmetries of the corresponding system of ODEs.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34M99 Ordinary differential equations in the complex domain
35B06 Symmetries, invariants, etc. in context of PDEs
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