Regularizing nonlinear Schrödinger equations through partial off-axis variations

We study a class of focusing nonlinear Schrödinger-type equations derived recently by Dumas, Lannes, and Szeftel within the mathematical description of high intensity laser beams. 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 $L^2$-supercritical situations, even in the case of only partial off-axis dependence. This provides an answer to an open problem posed by Dumas, Lannes, and Szeftel. Read More: https://epubs.siam.org/doi/10.1137/17M1131313