Partial Ly$\alpha$ thermalization in an analytic nonlinear diffusion model

During recombination, the cosmic background radiation is disturbed, in particular, by Lyman-alpha emissions from neutral hydrogen. It is proposed to account for the subsequent time-dependent partial thermalization of the Lyman-alpha energy content in an analytically solvable nonlinear diffusion model. The amplitude of the partially thermalized and redshifted Ly-$\alpha$ line is found to be too low to be visible in the cosmic microwave spectrum, in accordance with previous numerical models and Planck observations.