  We study the phase structure of the two-flavor Nambu--Jona-Lasinio (NJL)
model in the chiral limit, extending a previous study of the competition of an
inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase
[1, 2]. There, an analytic expression for the dispersion relations for
quasiparticle excitations in the presence of both a particular inhomogeneous
chiral condensate, the so-called chiral density wave (CDW), and a homogeneous
2SC condensate was found. In this work we show how to determine the dispersion
relations for arbitrary modulations of the chiral condensate in the presence of
a homogeneous 2SC condensate, if the dispersion relations in the absence of
color superconductivity are known. In our calculations, we employ two different
Ans\"atze for the inhomogeneous chiral condensate, the CDW as well as the
real-kink crystal (RKC). Depending on the value of the diquark coupling we find
a region of the phase diagram where the inhomogeneous chiral and the 2SC
condensates coexist, confirming results of Refs. [1, 2]. Decreasing the diquark
coupling favors the inhomogeneous phase over the coexistence phase. On the
other hand, increasing the diquark coupling leads to a larger 2SC phase, while
the inhomogeneous chiral and the coexistence phases become smaller. In
agreement with previous studies the RKC Ansatz is energetically preferred over
the CDW Ansatz. Both Ans\"atze lead to a qualitatively similar phase diagram,
however the coexistence phase is smaller for the RKC Ansatz.
