Solitons and their classical stability in complex scalar field theories with a global $U(1)$-symmetry

We consider different types of classical solutions arising in complex scalar field theories possessing a global $U(1)$-symmetry. Among them are extended axially-symmetric “Q-tubes” and nonlinear dips and rises in charged scalar condensate. The latter objects were not studied before in the relativistic field theory framework. We reveal main properties of these solutions and study their classical stability in theories allowing for analytical analysis. The role of unstable solutions in transitions between stable $Q$-balls and homogeneous condensate will be discussed. Also, linear perturbations on top of classically stable $Q$-balls will be studied in theories with the flat and power-like potentials.