We introduce a technique to obtain localization of Bose-Einstein condensates in optical lattices via boundary dissipations. Stationary and traveling localized states are generated by removing atoms at the optical lattice ends. Clear regimes of stretched-exponential decay for the number of atoms trapped in the lattice are identified. The phenomenon is universal and can also be observed in arrays of optical waveguides with mirrors at the system boundaries.

Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called mean-field $\mathit{XY}$ model and by the ${\displaystyle {\varphi }^4}$ lattice model, respectively. For both models the relationship between phase transitions and the topological properties of certain submanifolds of configuration space are exactly known. It turns out that these a priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space.

The dynamics of the three coupled bosonic wells (trimer) containing N bosons is investigated within a standard (mean-field) semiclassical picture based on the coherent-state method. Various periodic solutions (configured as π-like, dimerlike, and vortex states) representing collective modes are obtained analytically when the fixed points of trimer dynamics are identified on the $N=\mathrm{const}$ submanifold in the phase space. Hyperbolic, maximum and minimum points are recognized in the fixed-point set by studying the Hessian signature of the trimer Hamiltonian. The system dynamics in the neighborhood of periodic orbits (associated with fixed points) is studied via numeric integration of trimer motion equations, thus revealing a diffused chaotic behavior (not excluding the presence of regular orbits), macroscopic effects of population inversion, and self-trapping. In particular, the behavior of orbits with initial conditions close to the dimerlike periodic orbits shows how the self-trapping effect of dimerlike integrable subregimes is destroyed by the presence of chaos.

For physical systems described by smooth, finite-range, and confining microscopic interaction potentials $V$ with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that, unless the equipotential hypersurfaces of configuration space ${\Sigma }_v=\{(q_1,\dots ,q_N)\in {\mathbb{R}}^N|V(q_1,\dots ,q_N)=v\}$, $v\in \mathbb{R}$, change topology at some $v_c$ in a given interval $[v_0,v_1]$ of values $v$ of $V$, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature $\mathbf{(}\beta (v_0),\beta (v_1)\mathbf{)}$ also in the $N\to \infty$ limit. Thus, the occurrence of a phase transition at some ${\beta }_c=\beta (v_c)$ is necessarily the consequence of the loss of diffeomorphicity among the $\{{\Sigma }_v{\}}_{v<v_c}$ and the $\{{\Sigma }_v{\}}_{v>v_c}$, which is the consequence of the existence of critical points of $V$ on ${\Sigma }_{v=v_c}$, that is, points where $\nabla V=0$.