  We study a class of focusing nonlinear Schroedinger-type equations derived
recently by Dumas, Lannes and Szeftel within the mathematical description of
high intensity laser beams [7]. These equations incorporate the possibility of
a (partial) off-axis variation of the group velocity of such laser beams
through a second order partial differential operator acting in some, but not
necessarily all, spatial directions. We study the well-posedness theory for
such models and obtain a regularizing effect, even in the case of only partial
off-axis dependence. This provides an answer to an open problem posed in [7].
