For the number density of solvent molecules, nonlinear effects on solvation dynamics are studied using the dynamical density functional method. The present method includes nonlinear coupling between the number density and a polarization field only in the free energy functional. By means of the nonlinear free energy functional, nonlinear differential-integral equations are developed for the polarization field and number density. Numerical calculations show that solvent molecules relax more slowly around an ion than around a neutral solute. This result agrees qualitatively with nonlinear effects observed in many molecular dynamics simulations. In addition, the nonlinear dynamics of hydrogen bonds can be understood, by considering the slow relaxation of the number density of solvent molecules.