Focusing on a line for nonlinear Schrödinger equations in $\mathbb{R}^{2}$

Abstract

This paper is devoted to the study of a semi‐classical NLS equation with a small parameter ε in two space dimensions, with oscillating data that are highly oscillating in one direction only, with the aim of modelling geometric optics with a caustic consisting of a line in $\mathbb{R}^{2}$ . We prove that the phenomena encountered are typically one‐dimensional. In order to describe the results, we introduce spaces that make it possible to define scattering operators with a parameter.