Axion Gravitational Waves in the Presence of a Non-perturbative Hamiltonian

We study axion gravitational waves in the presence of a non-perturbative Hamiltonian. We find that the axion waves are described by the scalar fields of the graviton and the scalar fields of the axion. The presence of the non-perturbative Hamiltonian results in the propagation of axion gravitational waves. The propagation of the scalar fields of the axion is controlled by the polarization of the scalar waves. The propagation of the scalar fields of the axion is thus affected by the polarization of the scalar waves.