# Regularizing nonlinear Schroedinger equations through partial off-axis variations

## TL;DR

This paper investigates a class of focusing nonlinear Schrödinger equations with partial off-axis variations, establishing well-posedness and revealing a regularizing effect, thus addressing an open problem in the mathematical modeling of high intensity laser beams.

## Contribution

It introduces a well-posedness theory for nonlinear Schrödinger equations with partial off-axis variations, demonstrating a regularizing effect even with limited off-axis dependence.

## Key findings

- Proves well-posedness for the models
- Shows regularizing effect with partial off-axis variation
- Addresses an open problem in laser beam modeling

## Abstract

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].