Periodic oscillations of dark solitons in parabolic potentials

2007/05/08 by Dmitry E. Pelinovsky, D. E. Pelinovsky, P. G. Kevrekidis +2
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Nonlinear Photonic Systems #Other Condensed Matter (cond-mat.other) #cond-mat.other

paper · pdf · doi:10.48550/arxiv.0705.1016

20 pages, 3 figures

arxiv created 2007/05/08 · openalex publication_date 2007/05/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We reformulate the Gross-Pitaevskii equation with an external parabolic potential as a discrete dynamical system, by using the basis of Hermite functions. We consider small amplitude stationary solutions with a single node, called dark solitons, and examine their existence and linear stability. Furthermore, we prove the persistence of a periodic motion in a neighborhood of such solutions. Our results are corroborated by numerical computations elucidating the existence, linear stability and dynamics of the relevant solutions.

Related

• Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation
• On existence of dark solitons in cubic-quintic nonlinear Schrödinger equation with a periodic potential
• Orbital stability of the black soliton for the quintic Gross-Pitaevskii equation
• Stability of discrete dark solitons in nonlinear Schrodinger lattices
• Persistence of integrable wave dynamics in the Discrete Gross--Pitaevskii equation: the focusing case