Two-soliton molecule bouncing in a dipolar Bose–Einstein condensates under the effect of gravity

The dynamics of a two-soliton molecule bouncing on the reflecting atomic mirror under the effect of gravity have been studied by analytical and numerical methods. The analytical description is based on the variational approximation. In numerical simulations, we observe the resonance oscillations of the two-soliton’s center-of-mass position and width, induced by modulated atomic mirror. Theoretical predictions are verified by numerical simulations of the nonlocal Gross–Pitaevskii equation (GPE) and qualitative agreement between them is found. Hamiltonian dynamic system for a dipolar Bose–Einstein condensates (BECs) has been studied.

Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale𝕋, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for𝕋=ℝand𝕋=ℤwithin one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i)M(λ)theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.