Knot soliton solutions for the one-dimensional non-linear Schr\"{o}dinger equation

TL;DR

This paper reveals that certain breather solutions of the one-dimensional nonlinear Schrödinger equation can form knotted space curves, highlighting a topological aspect in a fundamental nonlinear wave model relevant to various physical systems.

Contribution

It demonstrates the existence of knotted soliton surface solutions in the nonlinear Schrödinger equation, introducing a novel topological perspective to this well-studied model.

Findings

Breather solutions can form knotted space curves.

Topological features are significant in a 1D nonlinear wave model.

Implications for physical systems modeled by the nonlinear Schrödinger equation.

Abstract

We identify that for a broad range of parameters a variant of the soliton solution of the one-dimensional non-linear Schr\"{odinger} equation, the {\it breather}, is distinct when one studies the associated space curve (or soliton surface), which in this case is knotted. The signi ficance of these solutions with such a hidden non-trivial topological element is pre-eminent on two counts: it is a one-dimensional model, and the no nlinear Schr\"{o}dinger equation is well known as a model for a variety of physical systems.