  The ground-state magnetic phase diagram of a spin $S=1/2$ two-leg ladder with
alternating rung exchange $J_{\perp}(n)=J_{\perp}[1 + (-1)^{n} \delta]$ is
studied using the analytical and numerical approaches. In the limit where the
rung exchange is dominant, we have mapped the model onto the effective quantum
sine-Gordon model with topological term and identified two quantum phase
transitions at magnetization equal to the half of saturation value from a
gapped to the gapless regime. These quantum transitions belong to the
universality class of the commensurate-incommensurate phase transition. We have
also shown that the magnetization curve of the system exhibits a plateau at
magnetization equal to the half of the saturation value. We also present a
detailed numerical analysis of the low energy excitation spectrum and the
ground state magnetic phase diagram of the ladder with rung-exchange
alternation using Lanczos method of numerical diagonalizations for ladders with
number of sites up to N=28. We have calculated numerically the magnetic field
dependence of the low-energy excitation spectrum, magnetization and the on-rung
spin-spin correlation function. We have also calculated the width of the
magnetization plateau and show that it scales as $\delta^{\nu}$, where critical
exponent varies from $\nu =0.87\pm0.01$ in the case of a ladder with isotropic
antiferromagnetic legs to $\nu =1.82\pm0.01 $ in the case of ladder with
ferromagnetic legs. Obtained numerical results are in an complete agreement
with estimations made within the continuum-limit approach.
