Due to experimental advances, the dynamics of strongly correlated systems is at the forefront of interest in contemporary research. In low-dimensional systems, quantum fluctuations are enhanced, and strong correlations often occur. Despite these complications, a number of such systems are integrable, which allows derivation of many exact results. In addition, there exist very powerful non-perturbative techniques which can be used even when integrability is broken. Recently there has been a tremendous progress in understanding equilibration and thermalisation in such systems, as well as the special equilibrium states (generalised Gibbs ensemble) that characterise the equilibrium state of integrable ones. However, much less is understood concerning the temporal unfolding of out-of-equilibrium dynamics, and the construction of correlation functions is also quite a challenge, both in and out of equilibrium. Our specific aims include - extend methods to simulate the time evolution to new systems (cf. e.g [1,2] and references therein); - finding an efficient construction for correlation functions in and out of equilibrium and use them to study the physical properties of these systems [3]; - develop hydrodynamical descriptions for experimentally relevant situations [4]; - understanding the effect of long-lived metastable excitations [5]. We shall apply existing, and also develop new analytic and numerical methods to investigate these issues, and to obtain a full quantum description of the dynamics both in the continuum and on the lattice. Besides solving theoretical problems, we also aim to obtain results that are directly relevant in the experimental context. References: [1] D.X. Horváth, K. Hódsági, and G. Takács: Non-equilibrium time evolution and rephasing in the quantum sine-Gordon model, Phys. Rev. A 100, 013613 (2019), arXiv:1809.06789 [cond-mat.quant-gas] [2] D.X. Horváth, M. Kormos, S. Sotiriadis, and G. Takács: Inhomogeneous quantum quenches in the sine-Gordon theory, arXiv:2109.06869 [cond-mat.str-el] [3] I. Kukuljan, S. Sotiriadis, and G. Takács: Correlation functions of the quantum sine-Gordon model in and out of equilibrium, Phys. Rev. Lett. 121, 110402 (2018), arXiv:1802.08696 [cond-mat.stat-mech] [4] B. Bertini, L. Piroli, and M. Kormos: Transport in the sine-Gordon field theory: From generalized hydrodynamics to semiclassics, Phys. Rev. B 100, 035108 (2019), arXiv:1904.02696 [cond-mat.stat-mech] [5] B.C. Nagy and G. Takács: Collapse instability and staccato decay of oscillons in various dimensions, Phys. Rev. D 104, 056033 (2021), arXiv:2105.01089 [hep-th]