Quasiperiodic oscillations and homoclinic orbits in the nonlinear nonlocal Schr\"odinger equation

TL;DR

This paper investigates the stability and dynamics of higher-order bright solitons in nonlinear nonlocal media, revealing the origins of quasiperiodic oscillations and shape-transformations through linear stability analysis and dynamical visualization.

Contribution

It provides a detailed linear stability analysis of higher-order solitons in the nonlinear nonlocal Schrödinger equation, linking unstable modes to observed oscillatory behaviors and shape changes.

Findings

Identification of unstable modes leading to quasiperiodic oscillations

Visualization of homoclinic orbits linked to shape-transformations

Linking dynamical states to observed soliton behaviors

Abstract

Quasiperiodic oscillations and shape-transformations of higher-order bright solitons in nonlinear nonlocal media have been frequently observed in recent years, however, the origin of these phenomena was never completely elucidated. In this paper, we perform a linear stability analysis of these higher-order solitons by solving the Bogoliubov-de Gennes equations. This enables us to understand the emergence of a new oscillatory state as a growing unstable mode of a higher-order soliton. Using dynamically important states as a basis, we provide low-dimensional visualizations of the dynamics and identify quasiperiodic and homoclinic orbits, linking the latter to shape-transformations.