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

Article type: Research Article

Authors: Carles, Rémi

Affiliations: IRMAR, Université de Rennes I, 35042 Rennes, France E‐mail: carles@maths.univ‐rennes1.fr

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.

Journal: Asymptotic Analysis, vol. 24, no. 3-4, pp. 255-276, 2000