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We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density, which is completely suppressed at the boundary position. Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations modify the density profile considerably. The local density approaches the average one as an inverse square of the distance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations from the boundary. We then consider the problem of localization of a foreign quantum particle (impurity) in the potential created by the inhomogeneous boson density. At the mean-field level, we find exact results for the energy spectrum of the bound states, the corresponding wave functions, and the condition for interaction-induced localization. The quantum contribution to the boson density gives rise to small corrections of the bound-state energy levels. However, it is fundamentally important for the existence of a long-range Casimir-like interaction between the impurity and the boundary.

In the presence of dipolar interactions, the excitation spectrum of a Bose gas can acquire a local minimum. The corresponding quasiparticles are known as rotons. They are gapped and do not decay at zero temperature. Here we study the decay of rotons in one-dimensional Bose gases at low temperatures. It predominantly occurs due to the backscattering of thermal phonons on rotons. The resulting rate scales with the third power of temperature and is inversely proportional to the sixth power of the roton gap near the solidification phase transition. The hydrodynamic approach used here enables us to find the decay rate for quasiparticles at practically any momenta, with minimal assumptions on the exact form of the interparticle interactions. Our results are an essential prerequisite for the description of all the dissipative phenomena in dipolar gases and have direct experimental relevance.

We study the classic problem of the capacitance of a circular parallel plate capacitor. At small separations between the plates, it was initially considered in the 19th century by Kirchhoff, who found the leading and the subleading term in the capacitance. Despite a large interest in the problem, almost 150 years later, only the second subleading term has been found analytically. Using the recent advances in the asymptotic analysis of Fredholm integral equations of the second kind with finite support, here we study the one governing the circular capacitor, known as the Love equation. We find analytically many subleading terms in the capacitance at small separations. We also calculate the asymptotic expansion at large separations, thus providing two simple expressions which practically describe the capacitance at all distances. The approach described here could be used to find exact analytical expansions for the capacitance to an arbitrary number of terms in regimes of both small and large separations.