Binocular rivalry is investigated in a continuum model of the primary visual cortex that includes neural excitation and inhibition, stimulus orientation preference, and spike-rate adaptation. Visual stimuli consisting of bars or edges result in localized states of neural activity described by solitons. Stability analysis shows binocular fusion gives way to binocular rivalry when the orientation difference between left-eye and right-eye stimuli destabilizes one or more solitons. The model yields conditions for binocular rivalry, and two types of competitive dynamics are found: either one soliton oscillates between two stimulus regions or two solitons fixed in position at the stimulus regions oscillate out of phase with each other.