  We investigate the critical properties of the $S=1/2$ bond-alternating spin
chain with a next-nearest-neighbor interaction in magnetic fields. By the
numerical calculation and the exact solution based on the effective
Hamiltonian, we show that there is a parameter region where the longitudinal
incommensurate spin correlation becomes dominant around the half-magnetization
of the saturation. Possible interpretations of our results are discussed. We
next investigate the effects of the interchain interaction ($J^{\prime}$). The
staggered susceptibility and the uniform magnetization are calculated by
combining the density-matrix renormalization group method with the interchain
mean-field theory. For the parameters where the dominant longitudinal
incommensurate spin correlation appears in the case $J^{\prime}=0$, the
staggered long-range order does not emerge up to a certain critical value of
$J^{\prime}$ around the half-magnetization of the saturation. We calculate the
static structure factor in such a parameter region. The size dependence of the
static structure factor at $k=2k_{\rm F}$ implies that the system has a
tendency to form an incommensurate long-range order around the
half-magnetization of the saturation. We discuss the recent experimental
results for the NMR relaxation rate in magnetic fields performed for
pentafluorophenyl nitronyl nitroxide.
