We investigate numerically and theoretically the propagation of beams of different input profiles in a highly nonlocal three-dimensional (3D) nematic liquid crystal (NLC) in the presence of an externally applied bias voltage. For the widely accepted scalar model of beam propagation in uniaxial NLCs, we obtain approximate analytical solutions in the form of 3D breathing solitons and find that the underlying induced potential (the induced change in the refractive index) must be periodic as well, with the same period as the breathers. We demonstrate that in 3D, the nonlocal response of the medium depends not only on the beam power but also on the beam intensity structure. We determine the fundamental shape-invariant soliton solution of the scalar model and check its stability in propagation.
