The anharmonic vibrational dynamics in nearly stoichiometric ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4+\ensuremath{\delta}}$ is studied by means of anelastic and ${}^{139}\mathrm{La}$ NQR relaxation. In the absorption component of the elastic susceptibility as well as in the nuclear relaxation rate a peak is detected as a function of temperature, and a relaxation time $\ensuremath{\tau}=1.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\mathrm{exp}[(2800\phantom{\rule{0ex}{0ex}}\mathrm{K})/T]$ s is derived. The relaxation processes are attributed to tilt motion of the ${\mathrm{CuO}}_{6}$ octahedra in double-well potentials, whose cooperative character increases the effective energy barrier to the observed value. The analysis of the relaxation mechanisms has been carried out by reducing the dynamics of the interacting octahedra to a one-dimensional equation of motion. The solitonlike solutions correspond to parallel walls separating domains of different tilt patterns and give rise to pseudodiffusive modes which appear as a central component in the spectral density of the motion of the octahedra. The tilt waves may be considered to correspond to the dynamical lattice stripes observed in La-based and Bi-based high-${T}_{\mathrm{c}}$ superconductors.