  In this paper we investigate the phase structure of a (1+1)-dimensional quark
model with four-quark interaction and in the presence of baryon ($\mu_B$),
isospin ($\mu_I$) and chiral isospin ($\mu_{I5}$) chemical potentials.
Spatially inhomogeneous chiral density wave (for chiral condensate) and single
wave (for charged pion condensate) approaches are used. It is established that
in the large-$N_c$ limit ($N_c$ is the number of colored quarks) there exists a
duality correspondence between the chiral symmetry breaking phase and the
charged pion condensation (PC) one. Moreover, it is shown that inhomogeneous
charged PC phase with nonzero baryon density is induced in the model by
arbitrary small values of the chemical potential $\mu_{I5}$ (for a rather large
region of $\mu_B$ and $\mu_I$).
