  The quark-meson model is often used as an effective low-energy model for QCD
to study the chiral transition at finite temperature $T$, baryon chemical
potential $\mu_B$, and isospin chemical potential $\mu_I$. The parameters of
the model are determined by matching the meson and quark masses, as well as the
pion decay constant to their physical values using the on-shell and modified
minimal subtraction schemes. In this paper, we study the possibility of
different phases at zero temperature. In particular, we investigate the
competition between an inhomogeneous chiral condensate and a pion condensate.
For the inhomogeneity, we use a chiral-density wave ansatz. For a sigma mass of
$600$ MeV, we find that an inhomogeneous chiral condensate exist only for pion
masses below approximately 37 MeV. We also show that due to our parameter
fixing, the onset of pion condensation takes place exactly at
$\mu_I={1\over2}m_{\pi}$ in accordance with exact results.
