  The phase diagram of an antiferromagnetic ladder with frustrating
next-nearest neighbor couplings along the legs is determined using numerical
methods (exact diagonalization and density-matrix renormalization group)
supplemented by strong-coupling and mean-field analysis. Interestingly, this
model displays remarkable features, bridging the physics of the J_1-J_2 chain
and of the unfrustated ladder. The phase diagram as a function of the
transverse coupling J_{\perp} and the frustration J_2 exhibits an Ising
transition between a columnar phase of dimers and the usual rung-singlet phase
of two-leg ladders. The transition is driven by resonating valence bond
fluctuations in the singlet sector while the triplet spin gap remains finite
across the transition. In addition, frustration brings incommensurability in
the real-space spin correlation functions, the onset of which evolves smoothly
from the J_1-J_2 chain value to zero in the large-J_{\perp} limit. The onset of
incommensurability in the spin structure-factor and in the dispersion relation
is also analyzed. The physics of the frustrated rung-singlet phase is well
understood using perturbative expansions and mean-field theories in the
large-J_{\perp} limit. Lastly, we discuss the effect of the non-trivial magnon
dispersion relation on the thermodynamical properties of the system. The
relation of this model and its physics to experimental observations on
compounds which are currently investigated, such as BiCu_2PO_6, is eventually
addressed.
