We calculate the chameleon field potential for ultracold neutrons, bouncing
on top of one or between two neutron mirrors in the gravitational field of the
Earth. For the resulting non--linear equations of motion we give approximate
analytical solutions and compare them with exact numerical ones for which we
propose the analytical fit. The obtained solutions may be used for the
quantitative analysis of contributions of a chameleon field to the transition
frequencies of quantum states of ultra-cold neutrons bound in the gravitational
field of the Earth.Comment: 11 pages, 4 figure
