We study the sine-Gordon (SG) model at finite densities of the topological charge and small SG interaction constant, related to the one-dimensional Hubbard model near half filling. Using the modified Wentzel-Kramers-Brillouin approach, we find that the spectrum of the Gaussian fluctuations around the classical solution reproduces the results of the Bethe ansatz studies. The modification of the collective coordinate method allows us to write down the action, free from infrared divergencies. The behavior of the density-type correlation functions is nontrivial and we demonstrate the existence of leading and subleading asymptotes. A consistent definition of the charge-raising operator is discussed. The superconducting-type correlations are shown to decrease slowly at small soliton densities, while the spectral weight of right (left) moving fermions is spread over neighboring (formula presented) harmonics.