  We discuss the problem of competition between a superconducting (SC) ordered
state with a charge density wave (CDW) state in stripe phases of high $T_c$
superconductors. We consider an effective model for each stripe motivated by
studies of spin-gapped electronic ladder systems. We analyze the problem of
dimensional crossover arising from inter-stripe SC and CDW couplings using
non-Abelian bosonization and renormalization group (RG) arguments to derive an
effective $O(4)$-symmetric nonlinear $\sigma$-model in $D=2+1$ for the case of
when both inter-stripe couplings are of equal magnitude as well as equally RG
relevant. By studying the effects of various symmetry lowering perturbations,
we determine the structure of the phase diagram and show that, in general, it
has a broad regime in which both orders coexist. The quantum and thermal
critical behavior is discussed in detail, and the phase coexistence region is
found to end at associated $T=0$ as well as $T>0$ tetracritical points. The
possible role of hedgehog topological excitations of the theory is considered
and argued to be RG irrelevant at the spatially anisotropic higher dimensional
low-energy fixed point theory. Our results are also relevant to the case of
competing N\'eel and valence bond solid (VBS) orders in quantum magnets on 2D
isotropic square as well as rectangular lattices interacting via nearest
neighbor Heisenberg exchange interactions.
