Exact Soliton-like Solutions of the Radial Gross-Pitaevskii Equation

Abstract

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e., radial) Gross- Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painlevé transcendent. In each case the potential can be chosen to be time-independent.

arXiv comment8 pages, 7 figures. Version 2: stability analysis of the dark solution

References in corpus (16)

• Dark solitons in atomic Bose-Einstein condensates: from theory to experiments
• Motion of dark solitons in trapped Bose-Einstein condensates
• Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities
• Localized nonlinear waves in systems with time- and space-modulated nonlinearities
• Ring Dark Solitons and Vortex Necklaces in Bose-Einstein Condensates
• On the stability of standing matter waves in a trap
• Adiabatic passage by light-induced potentials in molecules
• Time-averaged adiabatic ring potential for ultracold atoms
• Oblique dark solitons in supersonic flow of a Bose-Einstein condensate
• Normal Modes of a Vortex in a Trapped Bose-Einstein Condensate
• Tailoring of vibrational state populations with light-induced potentials in molecules
• A Smooth, Inductively Coupled Ring Trap for Atoms
• Generation and evolution of vortex-antivortex pairs in Bose-Einstein condensates
• Dynamics of ring dark solitons in Bose-Einstein condensates and nonlinear optics
• Adjustable microchip ring trap for cold atoms and molecules
• Tailoring of motional states in double-well potentials by time-dependent processes

Cited by in corpus (10)

• The Snake Instability of Ring Dark Solitons in Toroidally Trapped Bose-Einstein Condensates
• Stabilization of ring dark solitons in Bose-Einstein condensates
• Planar and Radial Kinks in Nonlinear Klein-Gordon Models: Existence, Stability and Dynamics
• Creation and revival of ring dark solitons in a toroidal Bose-Einstein condensate
• Dynamics of interacting dark soliton stripes
• Self-interference of a toroidal Bose-Einstein condensate
• Adiabatic invariant analysis of dark and dark-bright soliton stripes in two-dimensional Bose-Einstein condensates
• An improved discretization of Schrodinger-like radial equations
• Reversible decay of ring dark solitons
• Parametric triggering of vortices in toroidally trapped rotating Bose-Einstein condensates