The problem of the stability of solitons in second-harmonic-generating media with normal group-velocity dispersion (GVD) in the second-harmonic (SH) field, which is generic to available chi^(2) materials, is revisited. Using an iterative numerical scheme to construct stationary soliton solutions, and direct simulations to test their stability, we identify a full soliton-stability range in the space of the systems parameters, including the coefficient of the group-velocity-mismatch (GVM). The soliton stability is limited by an abrupt onset of growth of tails in the SH component, the relevant stability region being defined as that in which the energy loss to the tail generation is negligible under experimentally relevant conditions. We demonstrate that the stability domain can be readily expanded with the help of two management techniques (spatially periodic compensation of destabilizing effects) - the dispersion management (DM) and GVM management. In comparison with their counterparts in optical fibers, DM solitons in the chi^(2) medium feature very weak intrinsic oscillations.
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