  The ground state phase diagram of a spin $S=1/2$ $XXZ$ Heisenberg chain with
spatially modulated Dzyaloshinskii-Moriya (DM) interaction $ {\cal H}= \sum_n
J\left[\left(S^x_n S^x_{n+1} +S^y_n S^y_{n+1}+\Delta S^z_n
S^z_{n+1}\right)+(D_0+(-1)^n D_1)\left(S^x_n S^y_{n+1} -S^y_n S^{x}_{n+1}
\right) \right] $ is studied using the continuum-limit bosonization approach
and extensive density matrix renormalization group computations. It is shown
that the effective continuum-limit bosonized theory of the model is given by
the double frequency sine-Gordon model (DSG) where the frequences i.e. the
scaling dimensions of the two competing cosine perturbation terms depend on the
effective anisotropy parameter $\gamma^*=J\Delta /\sqrt{J^2+D_0^2+D_1^2}$.
Exploring the ground state properties of the DSG model we have shown that the
zero-temperature phase diagram contains the following four phases: (i) the
ferromagnetic phase at $\gamma^*<-1$; (ii) the gapless Luttinger-liquid (LL)
phase at $-1<\gamma^*< \gamma^*_{c1}=-1/\sqrt{2}$; (iii) the gapped composite
(C1) phase characterized by coexistence of the long-range-ordered (LRO)
dimerization pattern $\epsilon \sim (-1)^n (S_n S_{n+1})$ with the LRO
alternating spin chirality pattern $\kappa \sim
(-1)^{n}\left(S^{x}_{n}S^{y}_{n+1} -S^{y}_{n}S^{x}_{n+1} \right)$ at
$\gamma^{\ast}_{c1}<\gamma^{\ast} <\gamma^{\ast}_{c2}$; and (iv) at
$\gamma^{\ast} >\gamma^{\ast}_{c2}>1$ the gapped composite (C2) phase
characterized in addition to the coexisting spin dimerization and alternating
chirality patterns, by the presence of LRO antiferromagnetic order. The
transition from the LL to the C1 phase at $\gamma^*_{c1}$ belongs to the
Berezinskii-Kosterlitz-Thouless universality class, while the transition at
$\gamma^*_{c2}$ from C1 to C2 phase is of the Ising type.
