Two-dimensional toroidal breather solutions of the self-focusing nonlinear Schrödinger equation

Abstract

We introduce the ring-like toroidal breather solutions of the (2+1)-dimensional nonlinear Schrödinger (NLS) equation with Kerr nonlinearity. When the width of the toroidal pulse is much smaller than its radius, the (2+1)-dimensional NLS equation in cylindrical coordinates can be approximated by the standard (1+1)-dimensional NLS equation. Based on this NLS equation, we obtain the first-order toroidal breather solutions, and study their dynamic characteristics. In particular, we discover and present for the first time the toroidal Peregrine soliton. This research can be extended to other high-dimensional nonlinear equations and provides a practical method for the study of other high-dimensional nonlinear localized wave structures.