In this thesis, we work on three problems. In the first problem, we work on a local time-dependent potential in a one-dimensional interacting bath. This work is motivated by cold atomic experiments, where one can shake the ultracold systems by using a time-dependent potential. We use linear response theory, bosonization, and renormalization group to compute the energy deposition in the one-dimensional bath by the time-dependent potential. We find that the energy deposition both for weak and strong coupling regimes shows a power-law behavior on the frequency of oscillation, and shows a different power-law exponent in strong and weak coupling limit. In the second problem, we work on the dynamics of a mobile impurity in a two-leg bosonic ladder. The theoretical and experimental studies of a mobile impurity in a one-dimensional bath in context of ultracold systems motivate this work. We confine the impurity to move in one of the legs of the ladder. We combine bosonization, renormalization group, and density matrix renormalization group to understand the dynamics of the impurity. We compute the Green's function of the impurity both theoretically and numerically. For a small interaction between the impurity and the bath, we find theoretically that the Green's function of the impurity decays as a power-law, and our numerical results show a good agreement with analytical result. Furthermore, we also give an analytical result of the Green's function for an infinite interaction, we find again that the Green's function decays as a power-law, and shows a good agreement with numerical results. Moreover, finally we give a semi-analytical expression for the power-law exponent at zero momentum as a function of interaction, again in good agreement with the numerical results. In the third problem, we are again investigating the dynamics of a mobile impurity in a two-leg bosonic ladder, but in this case, the impurity also tunnels in both longitudinal and transverse directions. We use similar methods than for the second problem. We compute the Green's function in the bonding and the anti-bonding modes of the impurity. The Green's function in the anti-bonding mode decays as power-law at small momentum and small interaction, while the Green's function in the bonding decays exponentially, and our numerical results show a good agreement with analytical ones.