REGULARIZING NONLINEAR SCHRO ̈DINGER EQUATIONS THROUGH PARTIAL OFF-AXIS VARIATIONS

We study a class of focusing nonlinear Schr ̈odinger-type equations derived recently by Dumas, Lannes and Szeftel within the mathematical de- scription 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 investigate the initial value problem for such models and obtain global well-posedness in L2-supercritical situations, even in the case of only partial off-axis dependence. This provides an answer to an open problem posed in [7].