Relative simplicity of the atomic structure of carbon nanotubes being hollow cylinders with walls formed by rings of six carbon atoms (generally, the walls can be multilayered) enables the researchers to use this class of substances as model one to reveal the basic mechanisms of the dynamics of quasi-one—dimensional systems. The present work studies the nonlinear properties of carbon nanotubes with strong electron interactions described by the Hubbard Hamiltonian. A microscopic Hamiltonian describing electrons in carbon nanotubes with allowance for the electron mobility, Coulomb repulsion of electrons in one site of carbon nanotubes, and changes in spacing of the neighboring sites caused by acoustic oscillations is suggested. An effective nonlinear system of equations describing the dynamics of electron wave functions within the framework of the suggested Hamiltonian is derived. The existence of nonlinear stable periodic oscillations of electron wave functions in the examined model, in particular, corresponding to acoustic oscillations with different polarization states is established. The influence of the problem parameters on the character of nonlinear wave stability is revealed.