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Hidden symmetries and linearization of the modified Painlevé-Ince equation. (English) Zbl 0785.34014

Summary: The linearization and hidden symmetries of the modified Painlevé-Ince equation, \(y''+\sigma yy'+\beta y^ 3=0\), where \(\sigma\) and \(\beta\) are constants, are presented. The linearization of this equation by a nonlocal transformation yields a damped (stable) or growing (unstable) harmonic oscillator equation for \(\beta>0\). Hidden symmetries are analyzed by transforming the modified Painlevé-Ince equation to a third-order ordinary differential equation (ODE) which, in general, is invariant under a three-parameter group by a Riccati transformation. A type I hidden symmetry is found of a second-order ODE found from the third-order ODE where a symmetry is lost in the reduction of order by the nonnormal subgroup invariants. A type II hidden symmetry occurs in the third-order ODE because the symmetries of a second-order ODE, reduced from the third-order ODE by another set of normal subgroup invariants, are increased.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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References:

[1] DOI: 10.1080/16073606.1985.9631915 · Zbl 0618.34009
[2] DOI: 10.1063/1.528096 · Zbl 0785.34031
[3] DOI: 10.1088/0305-4470/20/2/014 · Zbl 0625.34044
[4] DOI: 10.1016/0022-247X(90)90244-A · Zbl 0719.34018
[5] DOI: 10.1088/0305-4470/25/21/018 · Zbl 0764.34003
[6] DOI: 10.1063/1.526766 · Zbl 0587.34004
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