We consider systems of ODEs which are associated with some physically significant examples: shallow water equilibrium solutions, travelling waves of the Harry Dym equation, a Lotka-Volterra system of competing species and the geodesic flow on the triaxial ellipsoid. The first three are shown to share the following properties: (i) they are hyperelliptically separable systems (HSS) and, after a suitable nonlinear time transformation, become algebraically completely integrable (ACI) and (ii) they are of the weak Painlevé type and become full Painlevé after the application of this transformation. The geodesic flow on the other hand, although it passes the usual Painlevé test, does not possess a full set of free constants and thus one may not conclude whether it has the Painlevé property or not. This system is also HSS and becomes ACI after the application of a suitable nonlinear time transformation. We also combine our geometric-analytical investigation with a numerical analysis of the system in the complex plane and show that there is perfect correspondence between the results of the two approaches. This correspondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transformations to complete integrability are not available.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)