Stability analysis of orbital modes for a generalized Lane-Emden equation

DOI10.1016/J.CNSNS.2018.08.001zbMATH Open1509.85001arXiv1805.07625MaRDI QIDQ6301824

Ronald L. Adams, Stefan C. Mancas, Haret C. Rosu

Publication date: 19 May 2018

Abstract: We present a stability analysis of the standard nonautonomous systems type for a recently introduced generalized Lane-Emden equation which is shown to explain the presence of some of the structures observed in the atomic spatial distributions of magnetically-trapped ultracold atomic clouds. A Lyapunov function is defined which helps us to prove that stable spatial structures in the atomic clouds exist only for the adiabatic index $gamma=1+1/n$ with even $n$. In the case when $n$ is odd we provide an instability result indicating the divergence of the density function for the atoms. Several numerical solutions, which according to our stability analysis are stable, are also presented.

Mathematics Subject Classification ID

Theoretical approximation in context of PDEs (35A35) Galactic and stellar structure (85A15) Orbital mechanics (70M20) Hyperbolicity, Lyapunov functions for infinite-dimensional dissipative dynamical systems (37L45) Hydrodynamic and hydromagnetic problems in astronomy and astrophysics (85A30) Stability and instability of geophysical and astrophysical flows (76E20) Nonautonomous smooth dynamical systems (37C60)

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