Solvable model of strings in a time-dependent plane-wave background

We investigate a string model defined by a special plane-wave metric ds(2) 2du dv - lambda(u)x(2) du(2) + dx(2) with lambda and k/u(2) = const > 0. This metric is a Penrose limit of some cosmological, Dp-brane and fundamental string backgrounds. Remarkably, in Rosen coordinates the metric has a 'null cosmology' interpretation with flat spatial sections and scale factor which is a power of the light-cone time u. We show that: (i) this spacetime is a Lorentzian homogeneous space. In particular, it admits a boost isometry u' = lu, v' = l(-1)v similar to Minkowski space. (ii) It is an exact solution of string theory when supplemented by a u-dependent dilaton such that the corresponding effective string coupling e(phi(u)) goes to zero at u = infinity and at the singularity u = 0, reducing back-reaction effects. (iii) The classical string equations in this background become linear in the light-cone gauge and can be solved explicitly in terms of Bessel's functions, and thus the string model can be directly quantized. This allows one to address the issue of singularity at the string-theory level. We examine the propagation of first-quantized point-particle and string modes in this time-dependent background. Using an analytic continuation prescription we argue that the string propagation through the singularity can be smooth.