On periodic solutions and their modulations for the Manakov system

Periodic solutions of the Manakov system are studied with the aim to establish links between recent results of Wright (Physica D 2013 {\bf 264} 1) and Kamchatnov (EPL 2013 $\mathbf{\text{103}}$ 60003), where the one-phase solutions have been obtained by different methods and with different parameterizations. Two types of waves, which in the context of dynamics of two-component Bose-Einstein condensates can be called {\it density} and {\it polarization} waves, are determined as particular solutions of the Manakov system. Whitham equations describing modulations of these two types of waves are obtained.