Nonlinear Dynamics

, Volume 91, Issue 3, pp 1733–1753 | Cite as

Stability of solitons in time-modulated two-dimensional lattices

  • Nir DrorEmail author
  • Boris A. Malomed
Original Paper


We develop stability analysis for matter-wave solitons in a two-dimensional (2D) Bose–Einstein condensate loaded in an optical lattice (OL), to which periodic time modulation is applied, in different forms. The stability is studied by dint of the variational approximation and systematic simulations. For solitons in the semi-infinite gap, well-defined stability patterns are produced under the action of the attractive nonlinearity, clearly exhibiting the presence of resonance frequencies. The analysis is reported for several time-modulation formats, including the case of in-phase modulations of both quasi-1D sublattices, which build the 2D square-shaped OL, and setups with asynchronous modulation of the sublattices. In particular, when the modulations of two sublattices are phase-shifted by \(\delta =\pi /2\), the stability map is not improved, as the originally well-structured stability pattern becomes fuzzy and the stability at high modulation frequencies is considerably reduced. Mixed results are obtained for anti-phase modulations of the sublattices (\(\delta =\pi \)), where extended stability regions are found for low modulation frequencies, but for high frequencies the stability is weakened. The analysis is also performed in the case of the repulsive nonlinearity, for solitons in the first finite bandgap. It is concluded that, even though stability regions may be found, distinct stability boundaries for the gap solitons cannot be identified clearly. Finally, the stability is also explored for vortex solitons of both the “square-shaped” and “rhombic” types (i.e., off- and on-site-centered ones).


Resonance Variational approximation Gap solitons Vortex solitons Bose–Einstein condensate 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

This article does not contain any studies with human or animal subjects.


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Authors and Affiliations

  1. 1.Department of Physical Electronics, School of Electrical Engineering, Faculty of EngineeringTel Aviv UniversityTel AvivIsrael
  2. 2.ITMO UniversitySt. PetersburgRussia

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